-- The MASS of each object will be the same after the collision as it was before it. But the velocities of both objects will be different before and after.
-- The momentum of each object is always (its mass) times (its velocity). Since the velocities change during the collision, the momenta will change.
BUT . . .
-- The SUM of the momenta of the two objects must be the same after the collision as it was was before.
That's what "conservation" means. No matter what happens to the two objects, no brand new momentum can appear from heaven, and none of their momentum can disappear. Any momentum that one object loses, the other object must gain, so that their SUM doesn't change.