To import compositions from one AE project into another you need to do the following:
We have two AE projects. Project 1 has a composition I want to open in Project 2.
1. In the project panel select the composition you wish to copy.
2. In project 1 go to File > Dependencies > Collect files.
3. In collect files panel make sure "For Select Comps" and "Reduce project" are selected. Hit collect.
4. Choose where to save (I save project 1 in the same folder as project 2 to keep it simple).
5. Open project 2. Go to File > Open Project. Doing so will automatically close Project 1.
6. Double click in the project panel of project 2, as you would to import any other file. Navigate to Project 1, double click it. Voila, composition from project 1 now available in project panel of project 2. Have fun!