One of the most well-known "offenses" is mentioned in Talmud Bavli Rosh Hashannah 25a. Summary:
Rav Gamliel, who was the Nassi ("chief") at the time and Rav Yehoshua had a disput regarding the validity of accepting a certain type of witness claiming the citing of the 1st crescent moon. Such citing affected the day of the Molad (in this case, it was Molad Tishrei), which, in turn, affects the day of Rosh Hashannah and Yom Kippur. Rav Gamliele challenged Rav Yehoshua to appear before him carrying his stick on the day that was Yom Kippur according to his (Rav Yehoshu'as) calculation (i.e. - Rav Gamliel challenged him to violate Yom Kippur based on R. Yehoshu'as calculation.) At the end, Rav Yehoshua did visit Rav Gamliel, and Rav Gamliel praised him. He acknowledged and even praised his scholarship (thus showing that he was not PERSONALLY offended by what he did) and also praised him for accepting his (Rav Gamliel's) ruling.
Briefly, Rav Gamliel did not take any personal offense. But he was concerned that there should not be different opinions regarding the molad and thus this would divide B'nai Yisra'el, and you would have different people celebrating Yom Kippur on different days! Yom Kippur, esp., is a day that demonstrates unity among Klal Yisra'el. Rav Gamliel wanted to be sure that everyone follows his opinion being Nassi. What makes his opinion "right" is a worthwhile discussion that you can find in the Mishnah / Gemarra pages, there.
Personal note: After reading this story, I understand a bit better our great loss in not having the Bet Hamikdash. Then, even with personal disagreements, they deferred to the opinion of the Nassi. Halevai, we could have a SINGLE world-wide "Va'ad Harabanim" whose opinion EVERYONE would follow automatically, even if we have our personal disagreements. To relate it to your question, the tanna'im and amoraim were concerned far more about communal unity than personal offense. If anything, I think that we have a bigger problem NOW than then.