So I guess that gets to a question I have about Meshtastic. Does it just send messages?

Yes, Meshtastic is text only.

In the case of your weather station, does it just send a
message that is formatted a certain way (say JSON) and then at the the other end does it just
accept that message and then do something with it? Or is there a way to send some
sort of data packets via the mesh network?

Meshtastic has support for telemetry/sensor data transmission.
It also supports arbitrary byte data payloads via Python API.

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