The implications of WeaklyUp are application specific. (After all, as you point out, it's configurable whether WeaklyUp is even allowed.) WeaklyUp is a situation that indicates that a new node is in sync with the leader but that all nodes have not yet agreed on the state of new node (because convergence hasn't been reached and therefore a true Up state obtained).
What you do with a node in that state is somewhat up to you. For example, a WeaklyUp node could likely receive incoming traffic from the outside world. Since the WeaklyUp node knows the complete state of the cluster it would know how to handle all incoming requests, even requests that require cluster-aware routing.
Akka Bootstrap (by default), for example, will report a WeaklyUp node as "ready" to Kubernetes. Thereby making the WeaklyUp node eligible to receive incoming traffic. So that's the canonical answer to your question: a WeaklyUp node can receive and process requests and route messages to other nodes in the cluster. It can also receive responses, of course, since remoting works as normal. Another example is that WeaklyUp members are also eligible for cluster-aware routing. Since if the cluster-aware router is aware of the WeaklyUp node, the fact that some other node may not be aware of the WeaklyUp node is moot.
However, as the docs indicate, there are a lot of limitations to WeaklyUp members because not all members have seen the new node and so it can't be allowed to do anything that would depend on convergence: like participating in quorum decisions or sharding.
