IMAP is the best solution, already mentioned. It might mean going to an external mail service, but that is no longer expensive, and many recommend it anyway to become ISP independent, and no longer have to change addresses in a panic if you migrate.
An alternative but workable testicle-aching procedure, (has that beaten the censor?), would be to track down where the OE message files are held, share it, and then from each machine when it wants to be up-to-date go in and copy them to it's own OE message directory.- it is about 6 layers deep but I do it often, as I copy them to another machine often for security - along with favorites. Takes a few minutes, but easy enough to automate with a simple BAT file and a desktop icon. It doesn't give you real-time copies, but once you do the BAT file it wouldn't be a problem to do it when required.