The Mishna, in Zevachim 9:3, lists blemished animals among those that are inherently invalid for the Altar, such that if they are brought up onto it, they are brought down again. Then, R' Chanina, Segan Hakohanim reports that
דּוֹחֶה הָיָה אַבָּא אֶת בַּעֲלֵי מוּמִין מֵעַל גַּבֵּי הַמִּזְבֵּחַ
My father used to push blemished animals off the altar.
R' Obadiah MiBartenura explains:
שאם עלו דוחה היה אותן כלאחר יד ולא היה מורידן דרך בזיון בפרהסיא
[Meaning] that if they ascended, he would push them away back-handedly and would not take them down in a degrading way, in public.
How would one get an animal off of the Altar without it being publicly noticed? The Altar was in the center of the Temple courtyard and the center of the action there. There was one access point - the open ramp on the south side. Generally, offerings would go up the ramp and be burned at the top, not come down, so I have trouble picturing how an animal could come down that way without being noticed.
If R' Chanina's father would literally push these animals off one of the other sides of the Altar, I would expect that such a dramatic decent would be even more noticeable, not to mention degrading. Perhaps he shoved them off the west side, where the only lines of sight available were from inside the courtyard, where only Kohanim were?