All sources are from Yeshiva.org.il's detailed article about this. Note that i am relying on them being correct -- i did not check them.
As we said in the question, there are two possible issues:
- Is it being warmed directly by the sun or not?
- Does the cold water get cooked upon entering?
It is most definitely a machloket about whether or not it can be used, so make sure to ask your rabbi before using it to get a personal psak.
Let's start with the first issue.
Some write that the water gets heated directly from the sun, and that the glass and pipes do not absorb enough heat to actually heat the water, making it permissible. These include the Tzitz Eliezer (7:19), Rav Moshe Tzvi Neriah, Rav Tzvi Pesach Frank, and Rav Shlomo Zalman Auerbach (הערות על ספר יסודי ישורון, עמוד לא; הו"ד בספר שולחן שלמה שיח ג יג).
On the other hand, Rav Binyamin Yehoshua Silber (שו"ת אז נדברו מלאכת אופה 13) writes that without the dud, the water would never heat up that much, and since it is the dud that makes the water get so hot (by focusing the sun's rays), it is considered toldat chamah, and is forbidden.
(Sorry, the article didn't bring Sephardic poskim on this matter.)
This is only regarding heating it initially. What about the second question, the new [cold] water that enters when you turn on the faucet?
Most poskim agree that this is a psik reisha d'lo neicha lei by an issur d'rabbanan.
Rav Ovadia Yosef writes (Yabia Omer part 4 OC 34) that it [a פרדלנ"ל by דרבנן] is permissible, but it is still good to be machmir, and if possible, stop the cold water from entering.
Ashkenazi poskim don't generally allow a פרדלנ"ל by a d'rabbanan, but this is a special case. We also have grama here -- you don't actually heat the cold water -- it happens on its own. As such, it's now a פרדלנ"ל plus two issurei d'rabbanan, and that is enough to be meikel and allow it, at least according to Rav Tzvi Pesach Frank (his shu"t OC 188), Harav Or Letzion [unsure who this is], and the Tzitz Eliezer (7:19, 20:16).
However, the Minchat Yitzchak (4:44) and Shevet Halevi (1:94) disagree with defining this as a פרדלנ"ל + two d'rabbanans, saying that this is not grama, and it is also a psik reisha d'neicha lei.
Shmirat Shabbat K'hilchata (1:45) uses the term "טוב להמנע" -- it is preferable to abstain. He also brings Rav Shlomo Zalman Auerbach that [despite his ruling that it is heated by the sun itself,] there are some systems that will automatically switch to using an electric boiler if there isn't enough sun, which is definitely forbidden, so it's better to avoid it.
The article sums it up: If one wants to be lenient, he has upon whom to rely, but as for one who is stringent -- תבוא עליו ברכה.