Mac OS X 10.0 only registers Services from aplications installed in /Applications, /Developer/Applications, or subfolders thereof, so if you want to use the Service provided by OmniDictionary you'll need to install it in one of those locations. Services can be provided by applications installed in /Network/Applications if the NSServicesFromNetworkApplications default (in NSGlobalDomain) is set to YES.
In order to keep my /Applications folder clean I had been placing apps that I had installed into a sub-folder of /Applications called User Applications. Once I moved my apps in /Applications I had several new entries in Services (OmniDictionary being one), woo-hoo!
The information on the OmniGroup web site implies that apps in subfolders of Applications would register as services (if applicable), but this doesn't seem to be the case.

