In P2P, a peer both provides and consumes services. A group of peers can then provide and consume services to and from each other without dependence on any one server. With this understanding, there’s an assumption that this consumption and distribution occurs when the peer is connected.
Within some P2P enabled applications, the communication may be cached or queued when the peer is not connected. I know this the way Groove works.
Within Freenet, any one of the nodes within the network can consume or supply files. But if a peer is not connected, it’s not part of the network, it isn’t a participant and files are consumed and supplied through other participants. Either you’re a peer, or you’re not. Again, the assumption of 24-hour access is not a factor.
Some systems support a hybrid cloud whereby service requests are cached at a remote location (usually hidden from the peer), waiting for the other peer to connect. When the other peer connects, the communication is concluded. The results of the service call can then be communicated back to the originating peer, or cached itself if the originating peer is offline.
In a true P2P system, any one of the peers within the network could act as a cloud (intermediary) for other peers. Within a hybrid system, such as Groove, the system itself might provide these types of intermediary services.
As for firewall issues, most P2P tools can work from within firewalls, or be made to work within firewalls.