After Sync Weave is installed on the second PC and you've specified the sync settings and told it to sync, you can go back to the first PC and select Tools → Weave → Sync Now.
Watch the bottom toolbar in Firefox. It will show what it's syncing. Bookmarks and toolbar bookmarks came first when we tried it. Mozilla warns that the first sync can takes a few hours to complete. For that reason, they invested the program with hueristics- routines that determine the "interestingness" of data to help it to decide what to sync first each time. As time goes by and we use the add-on, less data will be synced, and the sync will happen much faster.
It works! We synced the settings from a desktop PC to a new installation of Ubuntu 9.10 on a laptop, and Firefox there soon created the bookmarks. That was about an hour ago, and Firefox on the laptop has not yet obtained the "homepage" setting, but View → Sidebar → History already shows all the websites visited on the PC today and going back a couple of months. Also, entering a single character into the "Awesome Bar" brings up familiar URLs.
This is a very handy and promising utility in the Firefox ecosystem, but if you've visited any red light districts on your syncing PC, you might want to consider cleaning up the search history before that first sync.