Osmo is compatible with the ZAGG InvisibleShield for iPads, or any similar screen protectors that have the same or less thickness, and don't cover the edges of the device. Screen protectors that are thicker than the ZAGG InvisibleShield, or cover some of the device edges, are not compatible with Osmo.
Here are several other iPad screen protector brands that we recommend:
As for Fire tablets, we don't currently have a list of recommended brands. Any screen protector that is thinner than 0.3mm and has a cutout for the camera should work, but we can't guarantee that it will since we haven't tested any of them yet.