No, the full feature-set is available offline. You can browse your repos and any other repo you replicated, and push/pull. What offline means is simply that your changes won't immediately be replicated to peers. But the full app experience (Read and write) works offline.
You can't push within LAN and then have that automatically available when online to the wider network, unless you have some form of continuous replication to some public-facing host. This is what Radicle does for you.