Is there a linear space lower bound for streaming set equality?
There are both deterministic lower bounds and randomized upper bounds (for a version of the problem where you get check-ins and check-outs in a single stream rather than check-ins in one stream and check-outs in the other, but that makes little difference) in my paper

Space-efficient straggler identification in round-trip data streams via Newton's identitities and invertible Bloom filters.
D. Eppstein, and M. T. Goodrich.
arXiv:0704.3313.
IEEE Trans. Knowledge and Data Engineering 23 (2): 297–306, 2011.
(and WADS 2007).

See also:

What's the difference? Efficient set reconciliation without prior context.
D. Eppstein, M. T. Goodrich, F. Uyeda, and G. Varghese.
Proc. ACM SIGCOMM, Toronto, Canada, 2011.

which is more explicitly about comparing two sets.

In short: you can test whether two multisets (or sets with no repetitions) are the same using only very small space, or more generally find their symmetric difference in space roughly proportional to the size of the difference.

If you really mean sets, where repetitions are allowed, I think the problem may be much harder, though I don't know of a randomized lower bound for it.