Human beings are able to affect natural entities and humans are persons, so it seems clear that this is a logically possible kind of entity, because there are in fact such entities.  If God exists, then God would also be an example of an entity that is personal and that is able to affect natural entities.  The same is true of angels and ghosts and polytheistic gods. 

Hmmmm.  This seems a bit more puzzling. Ghosts are generally conceived of as able to affect natural entities (thus they rattle chains in attics, make stairs creak, make the air in a room turn cool, and cause images to appear in video and in photographs).  Those who believe in ghosts, believe that there is evidence of the presence and activity of ghosts, and this implies that ghosts are able to affect natural entities (chains, stairs, air temperature, and light).  But it seems like we can imagine a ghost-like person who was unable to affect natural entities.  A ghost who could not rattle chains, make stairs creak, cool the air in rooms, or even make any sort of visible appearance.  Obviously, such a ghost-like creature would not provide any empirical evidence of its existence, so we have no way of knowing (at least through our senses) that such persons exist, but it is not obvious to me that this idea is incoherent.  

A key objection to the idea of a ‘bodiless person’ is the probelm of personal identity for such ‘persons’.  Such a ghost-like person has no physical body (otherwise the physical body would allow the ghost-like person to affect natural entities), and furthermore such a ghost-like person could not communicate (in any ordinary fashion) what sort of memories, beliefs, and intentions it had, so we could not identify such persons by means of psychological criteria (such as possessing the same memories or the same personality or the same beliefs).  Nevertheless, even if we could not detect or know anything about the memories, beliefs, or intentions of such a person, it does not follow that such a person has no memories, no beliefs, and no intentions.  So, there could be continuity of memories over time in the person; it is just that we would not be able to know or determine this to be the case.

This is, no doubt, a controversial question (whether it is logically possible for ghost-like persons to exist who were unable to affect natural entities), but it is not clear to me, without further argument, that such persons are logically impossible.