It's been a long time since I've used Final Cut, so my apologies if this question has been asked and answered. (I searched the archive but didn't find what I was looking for.)
I thought that creating proxy media would have the side benefit of being able to take my library file - that has source media stored separately - onto another computer for editing. However, that doesn't seem to work and I just see "missing file" on the second computer.
Is there something I'm missing or is there not an easy way to work on a library without the source media?
You either need to modify your Library Storage Location settings before you make your proxy footage, or you didn't put FCPX into Proxy mode on the second computer.
The easiest way to do what you're wanting it to set up your project on your first computer. Import all of your footage. Change your Library Storage location for Media to In Library. Select all of your footage and transcode it to Proxy. Now all of your original media is stored and linked from where it originally was, but your Proxy media is stored in your library, so you can take just the library to another system, switch to Proxy mode, and edit away.