You can certainly edit on two monitors with Premiere Elements. In fact, under the View menu you'll find an option for dual-monitor set-up.
However, this set-up merely makes the Timeline panel a separate, floating panel. The rest of the interface remains in one block. So you can have your timeline in one panel as well as a couple of the floating panels, like the Audio Mixer and Clip Monitor. But the bulk of the program's interface can't be broken into separate panels.
It's always nice to have more desktop space. I feel claustrophobic with just one monitor!
Besides, I write books for programs like Premiere Elements, so I appreciate having my book text in one monitor and the program in the other.
It's nice that Premiere Elements can have its timeline separated from the rest of the interface. You'll especially appreciate it if you tend to use a lot of video tracks.
But only you can determine if a second monitor will make you more productive. In my opinion, $150 is a small investment for the extra room to spread out your work.