Presumably, the problem is that Mac OS X uses a revision of the AFP file-sharing protocol which is incompatible with the version used by the default Mac OS 7.6.1.
It turns out there is a fix for this issue:
- Install Open Transport 1.1.2 (a dependency for the next step) on the 7.6.1 Mac
- Install AppleShare Client 3.8.3 on the 7.6.1 Mac
The Apple article on AppleShare Client 3.8.3 has more information on this release of the software.
Apparently AppleShare 3.8.8 requires System 8.