Thank you for your post.
Internal networks are definitely easier to manage. Once you leave the internal network, you have no control how the outside network is going to perform, plus if the outside users don't have a static IP address, there will be another layer.
When a Find or Sort is performed from the client, the Index from the server is sent to the client. This allows the Find or Sort to execute quickly. Then, the record ID's are sent back to the server to obtain the information so it can then be displayed on the client machine.
A FileMaker Pro hosting company may or may not serve your remote clients faster than your server. The hosting company has a static IP address and probably has direct access to the internet (which your local network may not). You may want to contact a hosting company and ask them if you could test it out for a short period of time (4 hours or less). Doing so should help answer your questions.