There is currently no way of doing that (nor do I believe in any other editor), but obviously it would not be hard to do.
The main thing that would need thinking about is the file format (.osc files are specific to OSM data), the only thing that currently exist IMHO is the format used by the Notes dump on
planet.openstreetmap.org. To actually be useful at least JOSM would have to support it too.
Could you open an issue on github on the topic?
Note: you can naturally simply upload the notes to
osm.org or a private API instance.
Simon