No. Lightroom mobile is about sharing your Computer-based catalog with mobile devices and the web - not to other computers.
What you can do is use Lightroom Web via your browser on the other computer. You go into your browser and log into your Adobe account at https://lightroom.adobe.com/. You can then:
- use the P, U, X keyboard shortcuts to add pick flags
- use numbers 0-5 to add ratings
- remove photos from collections
- let others see a collection by making it public, sharing to FB etc, giving them the Adobe URL
- add new originals to LrMobile by dragging them from the other computer's folders and dropping them in the browser - the newly uploaded file will be automatically added to your main catalogue. Try this, it's got lots of potential uses.