At the very least, two people who are infested would be able to determine as much by contact. This is required by the Things' biology. Any part split off from a Thing is an independent living entity, a completely new organism. So an infected head and foot are capable of being independent organisms. In order to coordinate their actions (so as to pretend to be Palmer, for example), the adjacent Thing cells must be able to recognize each other and communicate. (And if Thing cells recognize that the tissue they are next to is not Thing, they will absorb and replace it.)
In the original story, Who Goes There?, the answer to this question is much clearer. The Thing in the story is telepathic. It learns to imitate people by reading their minds before consuming them. So in that case, the Thing would be able to identify any Thing-infected organism it encountered.