The meaning of "Rabbi Yossi would say"
In the daf for Shabbat, namely Menachot 13, going on to today’s daf, we encountered a particular mode of citation. That is, first there was a Mishnah, in which Rabbi Yossi took a position. Then, in the gemara immediately after the Mishnah, an Amora weighed in and explained Rabbi Yossi’s rationale / motivation.
On the first Mishnah of the perek, it is Reish Lakish who does this:
לְהַקְטִיר לְבוֹנָתָהּ לְמָחָר, רַבִּי יוֹסֵי אוֹמֵר: פָּסוּל וְאֵין בּוֹ כָּרֵת. אָמַר רֵישׁ לָקִישׁ: אוֹמֵר הָיָה רַבִּי יוֹסֵי, אֵין מַתִּיר מְפַגֵּל אֶת הַמַּתִּיר.
§ The mishna teaches that if one removed the handful from a meal offering with the intent to burn its frankincense on the next day, Rabbi Yosei says that the meal offering is unfit but partaking of it does not include liability to receive karet. Concerning this, Reish Lakish says: Rabbi Yosei would say, i.e., this is Rabbi Yosei’s reasoning: A permitting factor does not render another permitting factor piggul. In other words, if, while performing the rites of a permitting factor, one had intent to perform the rites of a different permitting factor outside its designated time, the offering is not rendered piggul on account of this intent.
וְכֵן אַתָּה אוֹמֵר בִּשְׁנֵי בְּזִיכֵי לְבוֹנָה שֶׁל לֶחֶם הַפָּנִים, שֶׁאֵין מַתִּיר מְפַגֵּל אֶת הַמַּתִּיר.
Reish Lakish adds: And you would say the same with regard to the two bowls of frankincense of the shewbread, that a permitting factor does not render another permitting factor piggul, and therefore if the priest burned one of the bowls with the intent to burn the other bowl the next day, the shewbread is not rendered piggul.
My assumption is that all of the words (even the ones where the English but not Hebrew gloss says “Raish Lakish adds) are all part of the quote being attributed to Rabbi Yossi.
On the next Mishnah, it is Rav Huna:
גְּמָ׳ אָמַר רַב הוּנָא: אוֹמֵר הָיָה רַבִּי יוֹסֵי, פִּיגֵּל בַּיָּרֵךְ שֶׁל יָמִין – לֹא נִתְפַּגֵּל הַיָּרֵךְ שֶׁל שְׂמֹאל..
GEMARA: Rav Huna says: Rabbi Yosei would say, in accordance with his opinion that intent of piggul with regard to one loaf or one arrangement does not render the second loaf or arrangement piggul, that if one had intent of piggul with regard to the right thigh, i.e., he slaughtered an offering with the intent to partake of the right thigh the next day, then the left thigh has not become piggul and one is not liable to receive karet for its consumption. What is the reason for this? If you wish, propose a logical argument, and if you wish, cite a verse.
In both cases, the statement is אוֹמֵר הָיָה רַבִּי יוֹסֵי, meaning “Rabbi Yossi would say”, or maybe indicating a frequency, “Rabbi Yossi was wont to say”.
Even so, I suspect that this phrasing, when used by Amoraim in this manner, is not necessarily intended as a true attribution and testimony as to what Rabbi Yossi literally, historically, said.
That is because of Rav Nachman (bar Yaakov)’s reaction to Rav Huna:
אֵיתִיבֵיהּ רַב נַחְמָן לְרַב הוּנָא: וַחֲכָמִים אוֹמְרִים, לְעוֹלָם אֵין בּוֹ כָּרֵת עַד שֶׁיְּפַגֵּל בִּשְׁתֵּיהֶן בִּכְזַיִת. בִּשְׁתֵּיהֶן – אִין, בְּאַחַת מֵהֶן – לָא.
Rav Naḥman raised an objection to Rav Huna from a baraita: And the Rabbis say that there is never liability to receive karet for partaking of the two loaves unless one has intent of piggul with regard to an olive-bulk of both of them, i.e., one’s intent renders both loaves piggul only if he slaughters the lambs with the intent to consume an amount equal to an olive-bulk from both loaves combined outside their proper time. One can infer from this that if he had intent with regard to both of them, yes, both loaves are piggul. But if his intent was with regard to only one of them, no, the other loaf is not piggul.
Note that we take out the words וַחֲכָמִים אוֹמְרִים, because the gemara proceeds to figure out the attribution, and suggests at first that it is Rabbi Yossi, then moves to Rabbi, and finally to the Sages. But, the objection stems from attributing it to Rabbi Yossi.
This objection is thus either directly from a contradictory brayta, or to an inference in a contradictory brayta, which we can figure out is Rabbi Yossi. Now, if Rav Huna literally had a tradition of a statement from Rabbi Yossi, that would seem to be stronger — even if it is not officially a brayta. But, what is a brayta if not an oral (rather than written) tradition of the words of Tannaim. It should withstand Rav Nachman’s brayta and derivation thereof. The same should apply to objections made by much later Amoraim, namely Rav Ashi and Ravina II.
It makes a lot more sense, then, if this is Rav Huna’s supposition of what Rabbi Yossi would have held. And the same for Reish Lakish — it was his supposition. And thus, the weird phrasing.