You're not paying the leasee but typically this would fall under either a capital lease or an operating lease based on a set of criteria, that I don't feel like typing out :). This would determine if you would need to depreciate the asset (capital lease only). Also, you would need to know what type of fixed asset it is, as land is not depreciable. Now that I think of it, if your not paying rent or anything for that matter why would you book a liability? You wouldn't. Very weird situation as it would never happen.

Fixed assets always depreciate regardless on what you're paying (except land as I noted above). The leasee would be booking the depreciation without a doubt because you are not paying anything.