What is your account type - POP or IMAP?

To answer the question as stated in your topic, it's possible but not easy.

Obviously you can enter the same filters by hand on each installation of Thunderbird, but it's tedious and error-prone. You can set them up on one machine and copy the filter files over to the other machines. Also tedious and error-prone.

The way you describe the messages vanishing from sight suggests strongly that you're currently using POP, and this has no way of making use of shared folders. To continue working in this way, all you could do is set each client to leave a copy on the server for the others and painstakingly manage a set of filters on each and every machine, keeping them in synch by hand and careful management. (And ditto for your folders.) You'd also have to attend to the POP server from time to time to remove accumulated messages left there by your clients.

The real answer to this and your later question is to use an IMAP-connected account, where all your data lives in one place, on the email provider's server, and all your devices simply look at and work with one common set of data. Any filtering, editing, moving, etc you did on one machine would very quickly become visible on the other machines. In fact. the ability for more than one machine to be simultaneously be working on the same message becomes a somewhat worrying possibility. It's not clear if this is a likely event in your scenario.

Even using IMAP, if you filter at the client then you'd still have the issue of managing multiple instances of your filters, and the final part of this is to make use of filtering offered by the server; then the filtering takes place "at source" and you have just one filter set to maintain.

If your current email provider doesn't provide IMAP, and cannot offer it, then there are many alternatives who will. If it's for business then I think you should look for a paid-for service that includes guarantees, backups and an SLA. Your domain provider should be able to offer this.

For personal use, many of the free suppliers are worth considering. Googlemail works well, but if you're uneasy about their pervasiveness, have a look at gmx/1&1. Microsoft's hotmail/live mail/outlook.com offers IMAP. Yahoo! probably does (please check in your locality) but they have a bad track record and a lamentable reputation and I wouldn't want to entrust them with any precious material.