I try to "offer keys" and "give keys", but no key is given even though the emotes say so.

Did you transfer toys from one HUD to another?

There is a bug in v4.13.00 HUD that causes the key to not be transferred. Subsequent versions should be ok (so update when available).

Workaround:

resetup the affected toy to get a new key.
- click the HUD Lulu logo, and on the menu, click "re-setup".
- after the setup, you need to give each owner a new set of keys.
OR, if you're the sub, look for the key in your inventory (you should have it when you ran setup last), and manually give this key to your owner.

This has been fixed in HUD v4.13.02 and above.

(Thanks to Mihli Daviau for reporting this, and Lieba Rabeni for the tests)