cats and dogs should only be allowed to sit up by their owner.

this is currently a way of griefing as someone can TRY to sit the cat and dog up, and it will still teleport to the owner.

this is very annoying.

I wouldn't say thats griefing. Its more like trolling.

I've never actually used cats/dogs. Maybe a pet flag or pet protection? No one can touch the pet?

a pet flag would not work as this is mainly a problem in the wild.

I am pretty sure this is already the case, when you tame them, they become yours.
I know I have seen dogs sitting at /waste and if you try to make them get up it say only the owner can do this.

I can say I've seen this too. So CooperCatCycling, did you actually sit up cats and dogs from other people yourself?

If you put cats and dogs on a lead connected to a fence post, they won't teleport to you, even if they're standing.

i remember a while back accidently clicking on one of kittens cats, and it teleported.

People have trolled me by tping my netherhounds to me :l