Open both documents in the same version.
@moogiemar – not a direct way and not copy/paste, but you could use Export to IDMS to create snippet files, that you can place. That is working in both directions. InDesign CC <=> InDesign 2014 (with the usual caveat of feature compatibility between the two versions).
Draging selected objects on the page to the file system will also create IDMS files. You can drag them from the file system into InDesign or use the Place command. The place gun will be populated and with the alt modifier key (Mac OSX) you could place the object(s) in place.
Or use Adobe Bridge to manage and place the IDMS files.
What's also possible:
Create and use a InDesign library file to store the objects and place the assets from that library.
When working with library files it would be best to create one from InDesign CC and one from InDesign 2014 for adding and removing assets. You can open the same library file in both InDesign versions. Library files are agnostic to InDesign versions.
I like to expand my suggestion for working with InDesign library files. One library created in InDesign CC, one library created in CC 2014. As already mentioned, you can have both libraries open in both versions. Drag and drop the objects you want to copy/paste to the appropriate library and drag them out to your page in whatever InDesign version you currently working:
The little lock icon at the bottom of the library will indicate that you cannot add items. But you can use the assets stored in the library.
Left: InDesign CC
Right: InDesign CC 2014
Of course opening a CC file in CC 2014 is possible and copy/paste from that. But you cannot open a CC 2014 file in CC. You first have to save a IDML file and open that in CC. Working with two libraries will be easier when you permanently like to exchange assets in both directions.
Thank you Willi and Laubender! Very helpful.