But do they share their mempool, when a node disconnects from the network for some time for example?
A node that disconnects from the network has to come back online, and reach the chain tip to start receiving unconfirmed transactions. Obviously they can't share the mempool with an offline computer.
Then these nodes communicate with its peers to check the validity of transaction on the mempool.
They communicate to exchange unconfirmed transactions. Not to check / question its validity. That happens locally, and is done by each node individually.