1. Both sides are false. If the first side is true then by the statement on the first side, the second side is true, and so by the statement on the second side the first side is false. Contradiction. Thus, the first side is false, and so the negation of the first side is true. The negation of the first side is that the second side is false. Therefore both sides are false.
2. Both sides are true. If either side is false, we have a contradiction to the statement on that side (assuming we are using the inclusive "or" as is the norm in math/logic). Conversely, if both sides are true, then there is no contradiction.
p.s. Have you ever read any of Ramond Smullyan's books? I'd recommend them if you like logic puzzles like this one.