In the case of a "club checkout" the pilot being "checked out" should log the hours IMHO.
The reason I say this is two fold: a) he is probably paying for them and b) insurance requirements may dictate he has been checked out and so he needs to log them.
For example my mate and me own an aeroplane. I have a lot more experience than him in complex fast aeroplanes and so our insurance company has put a restriction on him that he must complete 20 hours dual in the aircraft before he can solo. They say that these 20 hours dual can be with me or with a certified flight instructor (or CRI).
If it is with an FI then he can log P/UT and the FI P1, but he needs to log these hours as per the insurance requirement - else how do they know he has flown these 20 hours if an accident happens later on? If flying with me as safety pilot, then as he is rated to fly the aeroplane he can log them P1 and I'll be there "just in case" and log nothing.