I have a few teething toys here that I use if needed but I really haven't had to deal with them bringing their own (that I can remember lol). I'm fairly laid back about outside toys being brought in (it so rarely happens that it's not even an issue though) so I would likely allow them to bring one in. They all put toys in their mouth so that aspect doesn't bother me...just another toy to clean lol.

The amber teething necklaces....I actually bought one for a past child (remember my drooler??!?) and my last handful had one as well, it was totally fine and safe IMO. They are knotted before and after each bead so if it does break, it's not a huge hazard. I took it off my drooler for naptime because I wasn't sure what he'd do but the handful had had his on since just a few months old and his mom kept it on 24/7 and he never once played with it so I left his on. I have video monitors and would have been able to see if he got it off or Broke it though.