Scott was right; the person on the phone was wrong. The Mississauga Community Centres are terrific places, but their use is not free. It didn't seem possible, and indeed, it was not.
I visited the Misssissauga Valley Centre, the one closest to me. The pool is all right, not as nice as the one at the Y. Length swimming hours are very limited. The cost is $3.55 a pop, or a book of 10 swims with no expiration - that is, no expiry, in Canadian - for $32.00. It's a good deal, but the money I'd save isn't worth the restrictive hours.
Also, my membership at the Y includes everything. If I want to use the gym instead of swimming, I can. At the Community Centre, the gym, although inexpensive, is a separate charge.
I think I'll buy a book of 10 swims, to have another option in addition to the Y. If I used them over the course of a year or more, that would be fine. But I'll definitely keep my Y membership.