Gemara

(R. Zeira): The Korei (who calls the Kohanim or recites the words for them to repeat) may not call until the Tzibur finishes answering Amen (to Modim). The Kohanim may not bless until the Korei finishes. The Tzibur may not answer Amen until the Kohanim finish the Berachah. The Kohanim may not begin the next Berachah until the Tzibur finishes answering Amen to the previous Berachah.

39b (R. Zeira): When the Kohanim bless, the Tzibur says "bless Hash-m, mighty angels..."; "bless Hash-m, all legions..."; and "bless Hash-m, all His creations..."

40a (R. Chiya bar Aba): It is a mistake to say the verses outside the Mikdash.

Support (Rav Papa): Even in the Mikdash, it is not so proper. If a servant is being blessed, shouldn't he listen attentively?!

(Rav Acha bar Chanina): The verses should also be said outside the Mikdash. If a servant is being blessed, should he not show his appreciation?!

(R. Avahu): I used to say the verses. When I saw that R. Aba of Ako did not say them, I stopped saying them.

55b (Ameimar, Mar Zutra or Rav Ashi): If one saw a dream and does not know what he saw, he should say the following at the time of Birkas Kohanim:

"Hash-m. I am Yours, and my dreams are Yours... Whether I dreamed about myself or about others, or if they dreamed about me, if it was good, strengthen it... If it needs curing, cure it... Just like you reversed Bilam's (intended) curse to a blessing, reverse all of my dreams for my good."

Rishonim

Rif, Rosh and Rambam (Megilah 15b and 3:21,23 and Hilchos Tefilah 14:5): The Korei may not recite until the Tzibur finishes answering Amen. The Kohanim may not bless until the Korei finishes. The Tzibur may not answer Amen until the Kohanim finish the Berachah. The Kohanim may not begin the next Berachah until the Tzibur finishes answering Amen.

Rif and Rosh (ibid.): The Tzibur should not say the verses ("bless Hash-m...") outside the Mikdash. Even in the Mikdash, it is not so proper. If a servant is being blessed, shouldn't he show his appreciation?! (This is unlike our text.)

Rambam (14:8): The Kohanim do not begin to bless until the Shali'ach Tzibur calls 'Kohanim'. They answer "Yevarechecha", and he says every (subsequent) word before them.

Rosh (21): R. Tam says that the Shali'ach Tzibur may not call to the Kohanim. He may not interrupt Shemoneh Esre even for Amen Yehei Shmei Raba! Rather, the overseer of the Beis ha'Keneses calls them. It says 'the Kohanim may not bless until the Korei finishes', and 'the Shali'ach Tzibur may not begin...' because the Korei is not the Shali'ach Tzibur. Also R. Chananel and Medrashim say so. However, the Shali'ach Tzibur may say the words for the Kohanim to repeat.

Hagahos Maimoniyos (5): The people should not say the verses mentioned in Sotah. R. Chiya bar Aba said that it is a mistake to say them outside the Mikdash. At the time of Birkas Kohanim, my Rebbi would say "Hash-m, I am Yours...', like it says in Berachos.

Poskim

Shulchan Aruch (OC 128:18): The Korei may not recite until the Tzibur finishes answering Amen to Modim. The Kohanim may not begin 'Asher Kidshanu ...' until the Korei finishes. The Kohanim may not begin "Yevarechecha" until the entire Tzibur finishes answering Amen to Asher Kidshanu. They may not say a word before the one reciting it for them finishes. The Tzibur may not answer Amen until the Kohanim finish the Berachah.

Beis Yosef (DH Ein): The Tur explains that the Kohanim may not say a word of Birkas Kohanim before the one reciting it for them finishes. Rashi explains that they may not begin Birkas Kohanim until the one who calls them finishes. The Ran brings both explanations; perhaps both are true.

Question (Beis Yosef DH v'Ha): Why must we say that Kohanim may not begin a Berachah before the Tzibur finishes saying Amen? They must wait until the Korei finishes! According to the Rambam, we must teach about "Yevarechecha"; the Korei does not say this word for them. How can we answer for the Tur and Ran? Perhaps 'Kohanim may not begin' really means ' the one who is Korei for the Kohanim may not begin.' Or, in the days of the Gemara no one said the words before the Kohanim did. Or, Kohanim must wait for the Tzibur before saying 'Ribon ha'Olamim ...' (which they say after Birkas Kohanim).

Question (Bach DH u'Mah (2)): The Kohanim say 'Ribon...' after the Shali'ach Tzibur begins Sim Shalom, which is after the Tzibur finished saying Amen!

Answer (Taz 13): The Shali'ach Tzibur may begin once the majority finished. Kohanim must wait for all to say Amen, for Amen is part of their Berachah.

Rebuttal (Magen Avraham 28): This is unreasonable. It is more important that the Tzibur hear Sim Shalom! Rather, we can say that the Korei need not wait for people who delay Amen too long, but the Kohanim must wait for them.

Beis Yosef (DH v'Chosvu, citing Tosfos Sotah 39b DH Ad): One who blesses ha'Motzi need not wait for the minority who answer Amen too slowly, for they err. However, regarding Birkas Kohanim one must wait for them, for they must hear the Berachah. As long as they answer, they do not hear others.

Mishnah Berurah (64,65): The Korei needs to wait only for the majority of the Tzibur to answer Amen. The Kohanim must wait for everyone, because everyone must hear Birkas Kohanim.

Kaf ha'Chayim (106): Many say that also the Korei must wait for the entire Tzibur. One should be stringent about the Safek.

Kaf ha'Chayim (109): The Korei must also wait for the Tzibur to finish saying Baruch Hu u'Varuch Shemo (after Hash-m's name).

Mishnah Berurah (68): The Korei may not say a word before the Kohanim finish saying the previous word.

Shulchan Aruch (26): When the Kohanim bless, the people should not say any verse. Rather, they are silent and intend for the Berachah.

Taz (23, citing Tosfos 40a DH Kol): We permit one who saw a bad dream to say a Tefilah, for perhaps he needs to remedy the dream .

Rema: In any case, nowadays that Kohanim sing at length, the custom is to say verses. However, it is better not to say them.

Taz (25): One may say them while the Korei sings. The Gemara did not suggest this, for in those days he did not sing.

Mishnah Berurah (103): It is better not to do so. Certainly, one should not say them out loud. One who repeats the words of Birkas Kohanim errs.