Prerequisite
I will try to explain the problems. first some sources. Shulchan Aruch Choshen Mishpat, Hilchot Edut, 36, stated laws about numerous persons who did testimony and one of them was relative or unfit. But for a better understanding, we will begin from Mishna.
Mishna Maccot 1, 7-8:
עַל פִּי שְׁנַיִם עֵדִים אוֹ שְׁלשָׁה עֵדִים יוּמַת הַמֵּת (דברים יז),
אִם מִתְקַיֶּמֶת הָעֵדוּת בִּשְׁנַיִם, לָמָּה פָרַט הַכָּתוּב
בִּשְׁלשָׁה, ...
A verse says that the testimony recquires two or three witnesses, what must we learn from the word "three"?
מַה שְּׁנַיִם נִמְצָא אַחַד מֵהֶן קָרוֹב אוֹ פָסוּל עֵדוּתָן בְּטֵלָה
, אַף שְׁלשָׁה נִמְצָא אֶחָד מֵהֶן קָרוֹב אוֹ פָסוּל, עֵדוּתָן
בְּטֵלָה.
As well as if one of two witnesses is parent or unfit, the testimony is rejected, as well, if one of the three is, the testimony is rejected.
מִנַּיִן אֲפִלּוּ מֵאָה, תַּלְמוּד לוֹמַר, עֵדִים.
We learn from the verse that the rule is the same even if the witnesses are hundred and one of them is unfit or relative.
אָמַר
רַבִּי יוֹסֵי, בַּמֶּה דְבָרִים אֲמוּרִים, בְּדִינֵי נְפָשׁוֹת.
אֲבָל
בְּדִינֵי מָמוֹנוֹת, תִּתְקַיֵּם הָעֵדוּת בַּשְּׁאָר.
Rabbi Yosse limited this rule to cases of criminal affairs, no problem for business law. Rashi explained this following the general halachic tendency to searsh how to avoid death penalty. RI explained that in business one witness alone can change the din, contrarily to criminal affairs which need entire group of two witnesses, this is a sign of weakeness of criminal testimony.
רַבִּי אוֹמֵר,
אֶחָד דִּינֵי מָמוֹנוֹת וְאֶחָד דִּינֵי נְפָשׁוֹת.
Rabbi Yehuda Hanassi, rejected Rabbi Yose's restriction.
בִּזְמַן שֶׁהִתְרוּ
בָהֶן, אֲבָל בִּזְמַן שֶׁלֹּא הִתְרוּ בָהֶן, מַה יַּעֲשׂוּ שְׁנֵי
אַחִין שֶׁרָאוּ בְאֶחָד שֶׁהָרַג אֶת הַנָּפֶשׁ:
Therefore he defined a own restriction concerning the rule application, in cases in which the the Kasher and the unfit witnesses were intentionally together to alert the alleged perpetrator . E.G. if two brothers were witnesses of a assassination scene,, the perpetrator will never be condemned, despite other regular witnesses? If only one of the brothers made the alert (with a third person, also a witness but non relative), in this case his brother will not disturb the testimony. Rashi explained that this last rule is specific to criminal laws ( because the reference to alrrt which is necessary for criminal laws only). The gemara ruled as Rabbi Yehuda Hanassi. The Rosh in responsa 60, 5 said that to testimony is a duty in such a case.
Tosfot asked: For Rabbi Yossi, who doesn't agree with this last restriction, how can we validate testimony on Get, which is public, in presence of relatives? He answered in name of Rabbenu Itschaq that the rules of disqualification are only for witnesses who go to testimony in bet din (making punishment testimony {performative testimony}), but for their presence at time of the action only they don't disqualify. He rad th wordscof Rabbi Yehuda in the Mishna, as applyed in the case in which both brothers were comming in Bet Din. And tosfot adds in two places that even with a later appearence at court they disqualifie the testimony of Kasher witnesses.
Tosfot answered that the presence of unfit witnesses don't disturb the regular testimony if they are not associated to the fit witnesses on bet din ( or Perhaps on the Get signature).
Tosfot Yom Tov on the mishna said that the answer of Tosfot is not valuable for Rishonim who think that at time of seeing only they already cancel the future testimony of the Kasher witnesses ( Rashi, Rambam for instance).
He answered that if the relatives or infit witnesses didn't come toch kede dibbur after the Kasher group, they cannot disqualify the group ( against Tosfot opinion). For a testimony which is not linked to going at bet din, as Get testimony, Tosfot yom Tov answered for Rambam and Rashi and Bartenura, following the SMA that the pre-selection of witnesses at seeing time excludes all enlargement of the witnesses group. The unfit and relatives are excluded by pre-selection. For this novelty a source is needed, Perhaps, later we would discuss this.
In Gemara,
אמר רבא, הכי אמרי' להו: למיחזי אתיתו או לאסהודי אתיתו?
Judges ask them, why did you come there to see only or to testimony?
Coming where? Rashi, Ramban and Rosh said that judges ask unfit witnesses. Rabbenu Chayim Kohen said that they ask Kasher witnesses. The words of Rambam in chapter 5 of Edut are a condensation of the statement of Rabbi and the statement of Rava.
Tosfot explained that rava stated for business laws {I understand that Tosfot included with this words all testimony which needs no alert (Azhara) and Rambam also explains as this.
For Rabbi Yosse who ruled the rule of one witness who disqualify two good witnesses, in criminal law, without hatraa (the witness does not disturb if he did not alert), a relative can come and say that he was there as voluntary witness. Rav Chayim Kohen answered that we ask the good witnesses if they was associated with him. Tosfot Yom Tom answer that after kede dibbur he is not associated to the good group.
The Shach SA CM 36 sk 5 rejected the Rav Chayim Kohen's chiddush and regarded it as incongruent with most Rishonim. He answered that when the unfit witnesses are comming after the good testimony, they cannot cancel the good testimony, excepted cases in which their presence at the time of seeing fact was already known by Bet Din. So, separation of groups excludes problems of relative or unfit combination to the goog group as Tosfot Yom Tov.
The hearth of the matter
Now let's read poskim Acharonim. The Shach CM 36, end of sk 8 adressed the problem of the OP
לדידן העידנא פשיטא דכשר דהא מזמינים עדים כשרים מיוחדים לכך ואם כן פשיטא דהקרובים אינם מכוונים להעיד דאפילו יכוונו להעיד אינו פוסל דכיון דמזמינים עדים לכך הרי אנו מוציאים כל האחרים מכלל העדות ולאו כל כמיניהו דקרובים ופסולים להיות עדות בעל כרחנו וכך כתב מה"ר פרץ בהגהות סמ"ק סימן קפ"ג.
The Shach said generally relatives are not here to testimony because they know their unfitness. He adds that the witnesses must to be pre-selected, and through this pre-selection, we exclude all additional witnesses from the witnesses group. {accordin to this, the case cited in the answer of @sabbahillel needs explanations}. Mefarshim reported that Mahari Ben Lev treat the case in which a non pre-selected witness come spontaneously to testimony. see Ramban quoted in Shach bellow, it seems that a spontaneous intervention is not avoidable {may be against the opinion of Rav Chayim Kohen in tosfot quoted above, which give to the Kasher wtinesses the right to select with who they were.}.
Q:
do two kosher witnesses need to be uniquely specified as being
witnesses?
A --> Yes.
It is a good practice, lechatechilla. The advantage is to avoid addition of unfit witnesses or relatives (SMA, TYT and SHACH encourage this practice, in the further stuff, we see that Ramban himself report this practice)
Let's read further the Shach.
ועוד יש לומר דמסתמא לא חיישינן שכיוונו
להעיד והא דאמרינן בש"ס היכי אמרינן להו היינו כשבאין אחר כך מעצמן לבית
דין להעיד אבל כל זמן שלא באו לבית דין לא חיישינן להכי וכך כתב הרמב"ן
בחידושיו פרק יש נוחלין שם במסקנתו, וזה לשונו ומעשים בכל יום בקידושין
וכתובות כו' ואפשר לומר שכל זמן שלא באו לבית דין מסתמא הקרובים לא
כיוונו להעיד והתם מפני שבאו לבית דין הם צריכים לישאל אלא שאינו מנהג
אלא לזַמֵּן שם שנים כשרים להיות עדים ולא השאר עד כאן לשונו...
The question of the Gemara "how do we ask the witnesses?" is only for
witnesses who enter the Bet Din. The Ramban writted this in Chidushin
Baba Batra perek Yesh Nochalim. He concluded that without spontaneous intervention of relatives in Bet Din, we believe that relative persons were present only as not witnesses but spectators. Ramban also mentionned the practice of witnesses pre-selection.
Q
If no witnesses are specified, but there are certainly two (or
more) people that fit the description of a kosher witness and are
present, can this be taken as 'testimony' despite there being other
non-kosher witnesses (or invalid mixes of kosher witnesses, e.g.
relatives) witnessing this event?
A --> Yes
According to the first words of Ramban, we believe that the unfit and relatives were here to show only.
Q
are kosher witness-pairs within the crowd considered a valid
witnessing pair?
Shulchan Aruch Even Haezer 42, 4 in Rama.
ואפילו ייחד עדים יכולים אחרים שראו המעשה להעיד (ריב״ש סימן
תע״ט)
Despite that a first group of witnesses were selected, we can use a second group (from Tshuvot Rivash) but they need to see accurately the transaction.
See in the same Sayif also Pitche Teshuva EH 42, sk 11 in name of the Shut Chatam Sofer Even Haezer 100 mentioned by @Double AA a recomandation to pre-select witnesses for Kiddushin because for this kind of testimony there is no difference between to coming to look or to testimony. This tradition is reported in CS in name of Ritba. One of the Chidushim of this Chatam Sofer Responsa is that The Chatan an d Kalla are considered as they said that they want only Kasher witnesses to perform the Kinyan Kiddushin, and so they pre-exclude all to a unfit or relative witnesses, even if those witnesses was choose by the mesader Kiddushin. The Chatam Sofer make a difference in nama of the avne Miluim between Kidushin testimony and other testimonies. The Kiddushin tesimony is not only a testimony of the transaction but a part of it. He pointed that Tosfot in Makkot asked only about Gittin, not about Kiddushin.
A --> Yes
No mention is made that in a such case we need to ask them if they were intentionally wanted to testimony. They need to see the ring giving and hear the marriage statement "hare at mekudeshet li...". (I don't speak about an other kind of testimony, e.g. testimony that one woman is married). So the case of sabbahillel is understood.
