If you choose Typekit fonts, all of the users will have access and can add the same fonts.
If you are using other fonts, they can't be "synced." Theoretically, you would all have purchased licenses to the same fonts, and each user would have them installed on each computer.
However, if you package your fonts, when an InDesign document is opened on another computer, all of the fonts you don't have are made available as document fonts. Document fonts are only visible in that document, and can't be used in another file.
So is there a way I can "download" all the fonts from a users InDesign to upload onto all the other users computers?
If the fonts are not part of Typekit, you'd need to check the license for each font. Few fonts would permit you to share them with 10 users on one license. Each font has a separate license and may have different terms, so font housekeeping is a longwinded task. Hence the attraction of sticking to Typekit fonts.
In addition to the other responses in this thread, another solution would be use of a server-based, third party Font Management solution where each user's desktop system has client software that coordinates with the operating system and the Adobe CC applications. In this scenario, the fonts are installed on and accessed from the server.
However, this form of solution still requires that you buy the appropriate licenses to cover the total number of users you have.