I was always wondering this as well so thanks for bringing it up :)
It seems to be a machloket haposkim:
Per the comments we need to split the answer into two parts:
Opinions if any brachot are a hefsek at that point (someone davening with the tzibur and hears יראו עינינו) :
- In Sefer Ben Ish Chai פרשת פיקודי אות ה פרשה ראשונה he says to not answer amen beacause דאין אומרים אמן דברכות כאן. He is worried about a hefsek at that part.
- In Yechaveh Daat 2:30, Rav Ovadia says there is no hefsek to answer amen to brachot at that point.
- In the Kaf Hachaim O.C Siman 236 Seif Katan 21 he says,
דוקא בדיבור אסור להפסיק ביניהם אבל לענות קדיש וקדושה וברכו ואמן פשיטא דשרי דלא עדיף מק"ש וברכותיה.
"Specifically speaking one is not allowed to speak between the pesukim of יראו עינינו, but to answer kadish, kedusha, barchu, amen, it would for sure be allowed because it is no better than Shema and its brachot." Although he is talking about speaking in between the pesukim of יראו עינינו we can say the same thing after the pesukim of יראו עינינו that one can answer amen
- Mishneh Berurah O.C 236 Seif Katan 5 quoting the Tur says, ומ"מ אין להפסיק בדברים אחרים, meaning that things besides for יראו עינינו, would be a hefsek, but for יראו עינינו itself one can answer amen.
If there is a problem of answering amen to the actual bracha (someone just walking by and hears this brcha:
- In sefer מראה הבזק חלק ג שאלה טז in name of Rav Shaul Israeli he says:
וגם בן ארץ ישראל שנמצא בחו"ל, אף על פי שאינו נוהג לומר יראו עינינו, יענה אמן משום שענין סמיכת גאולה לתפילה ממילא לא נשמר מצד אמירת הקדיש. ומכיון שהללו האומרים את ברכת יראו עינינו שומרים על המנהג שנהגו, ובדין נהגו, כיון שמסיימם בברכה, גם אם אין הוא מתפלל הוא עמהם בעצם התפילה, יענה אמן.
"And also a person from the Land of Israel who is in chutz laretz, even though he does not customarily say יראו עינינו, he should respond Amen because the issue of linking geulah to teffilah is not affected by the recitation of Kaddish. And since those who recite the blessing of יראו עינינו are maintaining a custom that they have observed, and they are correct in doing so because they end with a blessing, even if he is not praying with them, by being with them in the prayer, therefore he should respond Amen."
Now although he was referring to Bnei Chutz laretz and not sefardim I think we can make the connection to sefardim, that if Jews in Eretz Yisrael who don't have a minhag to say יראו עינינו, say amen, then for sefardim as well who don't have the minhag also can answer amen.
- לע״ד it seems that according to Shulchan Aruch O.C Siman 236:2, there wouldn't be a problem/issur of saying יראו עינינו, and therefore no problem of answering amen as well.
אין לספר בין גאולה דערבית לתפלה ואף הנוהגין לומר י"ח פסוקים ויראו עינינו אין להפסיק בין יראו עינינו לתפלה ומיהו מה שמכריז ש"צ ר"ח בין קדיש לתפלת ערבית לא הוי הפסק כיון שהוא צורך תפלה וכן יכול לומר ברכו להוציא מי שלא שמע ולא הוי הפסק
"One should not talk between the blessing about redemption (ge'ulah) in the evening and the Amidah; even those who routinely say 18 verses and "yir'u eineinu" should not pause between that paragraph and the Amidah. What about when the reader announces Rosh Chodesh [so people can remember to say ya'aleh ve'yavo] between Kaddish and the evening Amidah? This is not a pause, since it is a requirement for the Amidah; and thus, one could also say that Barchu, to exempt one who did not hear it, wouldn't be considered a hefsek either."
Edit: Recently I have received an answer from Rabbi Yonatan Nacson who said that he has confirmed with Hagoan HaRav Yitzchak Yosef and Hagoan Rav David Yosef that one can answer amen to יראו עינינו. However Rav Ovadia in Moar Yisrael Volume 1 page 7 did not answer amen when he was praying in a minyan that recited this berachah (on one of his trips to South America), since it is not an obligation to answer amen to this berachah, but one may answer if he wants to.