Hi there.
It can be done (in theory) if there is someone in your city using them at the same time. But keep in mind there is the possibility that your scheduals could change so that you were both wanting them on the same day (which would not work because you need to get them cold enough beforehand.)
So if you feel assertive enough to say "actually i need to postpone my treatment by a day" if the need arises, then it could work. The hospitals are actually quite used to shuffling the scheduals around for various medical reasons, so it might not be as daunting or impossible as it might seem. Also a good idea to get a couple of contact numbers for the family or friends of the person you are sharing with, in case they become hard to contact (which can happen when chemo bleariness kicks in.)
I suggest you have a chat to the lovely Vanessa at Penguin Cold Caps NZ (who used them herself for her own chemo) Ph 0011 64 333 96 351.
I'm not sure what citiy you are in but I think the Epworth in Melb recently got some machines that do it (rather than individual caps you have to keep replacing with cold ones.) not sure what the cost implications of changing hospitals would be.