You should consider all the information you have, from all the (randomly allocated, and so informationally symmetric) thermometers. It doesn’t matter who was handed which thermometer. Forming accurate beliefs is normally about this simple. If you want the most accurate beliefs you can get, you’ll need to pay attention to the evidence. All of the evidence. Evenly.

This gives the impression that you think that normally one can just collect all the relevant evidence, after which you don't need to consider anyone else's opinion. I suppose it depends on what sort of world you live in, but that seems far from the normal situation to me.