Switching from Campy 53/39 to a 50/34. Am I going to need to remove some chain links?
 

If I do I will plan to buy the not very cheap Campy chain tool- just because I don't want to screw it up.

Check out Park tools.

Probably.

If it was my bike I'd definitely test the chain length and resize it if necessary. Assuming you're using a short arm rear derailleur you might find you don't have anough slack take up if you leave your chain as it is.

this a 20 speed drive train? maybe put a larger low gear cluster on back too
the 29 tooth cassette , so you still total the same for big + big tooth count.
-3 on front, + 3 on back..

It sort of depends on how you've got the chain sized now.

No sense shortening it until you know for sure. Put on the new crank with your current chain, put it in small-small, and see if the derailleur wraps it all up; if not, take out a link.

Buy the chain tool anyway.

That'll keep the big/big the same as it is now but it won't do anything for the little/little. If his slack take up is marginal now it won't work with the compact crankset unless he shortens his chain.

Yup. but chain length change was the question.

Chain wrap take up is another issue, but I expect the OP appreciates the
added issue of needing take up more wrap...
as spare parts , medium cage plates swap will be on the horizon..

I think that you're making this a lot more complicated than it needs to be. DiabloScott had the best answer. If the derailleur will take up the chain slack in the little/little, OP doesn't have to do anything. If it doesn't, OP simply needs to remove 1" of chain.

You don't need the Campy chain tool, read the info at Park Tool.