To give you an idea of the amount of reptiles in my house. There is 55 snakes and 2 lizard, well the blue tongue Skink died recently so just one bearded dragon now. I’ve had reptiles for 15 years and birds for 11 without issue and just having some basic hygiene when it comes to care/handling. I’ve touched one and then the other before but usually I do clean my hands between lizard and bird. My biggest Retic is 13ft so I have plenty of dangerous in the house. You just have to be smart if you have both out at the same time. And if you have one person holding the lizard and one with the bird it shouldn’t be a huge deal. But skinks are quicker than one would think and their jaws are decently powerful even though killing usually requires some shaking/thrashing for them.
I guess I just find it laughable someone having trouble controlling an aggressive bearded dragon when I control the Retics easily enough when they are in a grumpy mood. I obviously don’t have the big snakes out at the same time as the birds because the hunting mode on those things is crazy and their power is enough a strike will probably kill any of the birds without even coiling.
I would be cautious if you are feeding crickets to the lizards. The parasite load is very high for them so passing that between them would be my major worry. Their are plenty of way better more nutritious options though when it comes to feeders for lizards too.
Also, quarantine is a thing and I would make sure the animal is healthy itself before touching any other animal in the house.