I have my CFI right now, and I'm flying with a CFII in preparation for
my CFII checkride. Do I log that time as Dual Received, as well as PIC,
or just PIC, or just Dual Received?

Are you getting or giving instruction? If you are getting instruction,
I think you should be able to log it the same way as a non-CFI getting
instruction - that is, as dual received, and (for the time you are sole
manipulator) as PIC.

While we're on the subject, how should I log my students? I have one
who has his private, and is just flying with me to build time for his
instrument rating. Should he log our time together as both PIC and
Dual, or just Dual?

If a student has their PPL, they can log PIC when they are sole
manipulator, and dual when they are receiving instruction. This often
overlaps.

I have always thought you either logged PIC or
Dual, never both; but lately I've noticed others doing both, so I
thought I'd just ask.

One can log both.

By the way, if it means anything, I instruct at a Part 141 school.

Now that I've shot my mouth off, I don't know if 141 makes a difference
here. My comments are my understanding of part 91. (or is it 93?)

Jose
--
"Never trust anything that can think for itself, if you can't see where
it keeps its brain." (chapter 10 of book 3 - Harry Potter).
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