Technically it is possible to connect two instances of Klok to the same data file by storing the data file on a network share (or using something like Dropbox). However, there are some potential issues to watch out for which is why we don't "officially" support this yet.
If both people have the console open at the same time you might run into issues where one user doesn't see the effects that the other user has on the data. For example, if you create new projects or import time entries, the other user won't see those effects until they close and restart the console. Given that only one of you is writing to the database and the other is only reading from it, then this might not cause too much of a problem for you if you can coordinate this with the other user.
If you decide you want to try this, you can click on the Klokwork logo to access the preferences. In the connection settings you can see where the datafile is being store. You can change that to a network share and copy your existing data to that location. Then on the other user's machine, point it at the same location.