The idea of cryptographic hashes is that they don't leak information about the plaintext, though.
No, that's incorrect. A hash function should have these properties:
* Given a hash h it should be hard to find a message m such that hash(m) = h
* Given a mesage m1 it should be hard to find m2 such that hash(m1) = hash(m2)
* It should be hard to find any two messages m1 and m2 such that hash(m1) = hash(m2)
If you have some secure hash function h, the function h'(m) that appends the low byte of m to h(m) is still secure under all three properties, but it obviously leaks information.
Right, but you still include the sha-1 of the plaintext in your outgoing message, which is (IIRC) generally considered bad practice because it leaks information about the plaintext.