can both computers access the same database at the same time with a single license installation?
You cannot do this with a single license key. You'd need to purchase a second license from FileMaker.
If a second license is purchased then does Filemaker then work in a peer to peer type situation? Will a second license require a FM Server? Can a second license be purchased at a significantly lesser cost since a second software copy is not needed?
You'll need to talk directly to FileMaker about pricing. To acquire just two licenses, I would not expect much if any price break on the second one, but that's their call not mine.
FileMaker Pro can host up to 9 FileMaker Pro clients--each with a separate license key or all part of a "volume" license key.
A license number is equivilant to a full copy of Filemaker.
You could turn on Instant Web Publishing and access the database in a browser on computer B as long as it is running in FMP on computer A.
The client software is free on iPads and iPhones.
Custom Web Publishing would also allow access through a web browser but is very complicated.
Google "IWP site:forums.filemaker.com" https://www.google.com/search?q=+iwp+site%3Aforums.filemaker.com
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Thanks to both parties that helped me on this issue.