The private transfer map for Illinois indicates "Banned at gun shows, legal for long guns only elsewhere".

This is just not correct.

There is no difference in the legality of private transfers between long guns and handguns, or whether it is done at a gun show or not.

It is true that transfers that involve a gun show require an extra step of calling in for a NICS check, but that is the only difference between a private transfer involving a gun show and one that does not involve a gun show.

In all cases, the transferee must have a FOID card, and the waiting period applies.