This is a question I was asked on Quora, and it's a good one. It does happen from time to time.
It depends what kind of hearing aids they are. It’s not harmful to do this, and people are forced to wear mismatching aids due to losing one, or having their hearing change dramatically on one side.
For more advanced hearing aids, the pair communicate with each other (called Ear to Ear or E2E) for certain functions. Some of these functions are useful but don’t affect hearing ability - for example turning both aids up when you press the button on only one aid.
Some functions really do impact hearing ability though. One such function is where the pair of aids figure out which side is getting the clearest signal in a noisy place, then copying it across to the other aid. Adaptive directional microphones can work together, listening to the environment and then changing the direction in which the microphones “look” for sound. With RIC (receiver in canal) aids, you can have an array of 4 (or even 6!) microphones which pick up sounds from every direction. If they can’t communicate with each other through E2E they are limited in how well they can coordinate changing direction so you won’t be able to hear as well.
Likewise if you have a mobile phone, some aids use E2E to provide the caller’s voice in both ears simultaneously. Your phone connects to one hearing aid, and that aid sends the phone call to the other side so you hear it in stereo.
So, yes you can have mismatching aids but it’s better to have them paired up properly if they have advanced features. This is also the reason why those "invisible" aids are not great, because they don't communicate with each other and they only have one feeble fixed-direction microphone each!
And… always insure private hearing aids with the insurer agreeing to replace the pair if one is lost, so you don’t get stuck in this situation!