When a new node connects to the network, other nodes start to share new transactions with this new node they got. But do they share their mempool, when a node disconnects from the network for some time for example? I think that the mempool size is too large and it would be difficult to share their mempool state, but maybe I'm wrong...
A particular full node has its own subjective view regarding what to consider valid in the blockchain: it receives pieces of information from other nodes, verifies if everything is okay, and adds transactions to its local temporary storage of yet-to-be-confirmed financial transactions. Mempool is a completely optional feature: you, as an operator of your personal node, can decide whether to allocate memory for those unconfirmed transactions or not. The only drawback of you not having a mempool is that you can't tell other nodes about unconfirmed transactions, and you even can't see them until they get into the mined block. Some will say that without mempool, you will have an incomplete picture of the blockchain, and your node should no longer be considered "full" because you can't conduct full verification.