When I was in 5th grade, I walked to school. Our dog got out of the yard, and it was impossible to get him back in until he was ready, and I didn't have time to do anything about it. He followed me all the way to school, and ran across a the main road in front of the school, and it was my teacher he ran in front of. She slammed on her brakes, and he trotted over near me. That teacher and I didn't get along at all, and I was worried about her being in a mood because of the dog. Oh well.
As a teen, I had a cat that I used to play with, and my arms were always scratched up from his claws. That was a bit embarrassing.

Pets are pets. They're animals and they have no internal censor that tells them it's inappropriate to do certain things in "public." Do I get embarrassed when my cat runs around the apt with poop stuck to his fur? Of course not. I get a bit grossed out, of course, but it's not his fault. When I come home from work - a friend in tow - to find poop or streaks on the carpet ... well, that can be a little uncomfortable. But my friends are all animal lovers, and they understand. I even have one friend who has helped me clean up the mess on more than one occasion!

A farting dog could be embarrassing, but only if your friends think it was YOU and not the dog. My sister's dog has gas all the time, but everyone has learned to distinguish dog smell from people smell. So if it happens, we all just groan and spray some air freshener ... or, when it's really bad, we abandon the room.

Of course, when her younger dog would hump the older dog after a walk, well, that could sometimes be over the top. But it was funny more than embarrassing.