Every time I try to private trade with my friend, one of the pets seems to get lost and we never get it. We have to try and track it down. I'm worried that one of these times we won't get it back.

All transactions like that are finalized in one hour max, so there's no need to worry. Usually it takes just a few minutes. For an instant fix, please go to your new pet's profile using a link from the trade message, and update the pet's group manually.

Please check the list of known bugs in the sticked thread, this case id described there:

https://beastkeeper.com/thread/1226