Based on the description, you don't need both machines on at the same time. You do need to install Google Desktop on both machines though and only the files that are indexed after enabling "share across computers" will be available. The files are uploaded to a server linked to your Google account, so as long as the Google Desktop servers are running you will be able to access the files.
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In order to share your indexed files between your computers, we first copy this content to Google Desktop servers located at Google. This is necessary, for example, if one of your computers is turned off or otherwise offline when new or updated items are indexed on another of your machines. We store this data temporarily on Google Desktop servers and automatically delete older flies, and your data is never accessible by anyone doing a Google search. You can learn more by reading the Google Desktop privacy policy.
While your data is automatically deleted from our servers, you can use the Clear my Files from Google button to manually remove all your files from Google Desktop servers. Note that if these files haven’t yet been copied to your other computers, clicking this button will prevent you from finding them when you search from your other computers. The files will, of course, still be searchable from their computer of origin.
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