I think you should consider your possibilities.
The iOS iMovie version is greatly limited in capacity, not to mention disk space, compared to its sister application.
You can share 'projects' via iMovie theater, however I use that term loosely. Going back to practicality, if you share a complete project and sync it to your mobile device and lets say this mobile device only has 16GB HDD and you only have a 5GB iCloud storage locker than you will quickly run out of room in easily half of 1 project. With that being said, going back to the mobile devices capabilities, it doesn't have all the features your desktop version has so some things get lost in translation, you may lose a transition you set prior, etc. However, it is a resource to share projects nonetheless. Additionally, the main draw back of iMovie Theater is that it doesn't share the PROJECT thats why I was using that term 'loosely', it shares the entire project movie with it's components (i.e., background music etc) that you have created thus far or the selection of the project you dictate. Theoretically, you can download the iMovie project from the theatre to your device and create a new project on your iPhone where you can add clips and edit there. But you won't be able to edit the properties in the previous project.
What I recommend you doing if your going full scale with this, two of the better options I would recommend is :
- Get a Time Capsule, you can set it up to wake over WLAN and essentially use it as both a router and external HDD which files you can access from anywhere. Although, this isn't an iOS friendly option, it's a logical option, because why waste your time creating on an iOS when the features are limited and the projects are decapitated from your devices hardware.
- Dropbox, Box, Google Drive etc etc etc. Use a file sharing cloud service to where you can simply save the project there from the get go and edit through that file instead of any of your devices HDD (unless your planning on doing some high processing features in FCPX with rendering etc) then that is a viable option.
- Don't use iMovie. It's great and all, but look at its target audience, it's meant for the people that don't have much expertise or the attention span to sit and edit videos, it's one-click and done type editing.
- Just get a laptop and save yourself a lot of heartache.
But to answer your question specifically and without bias, then NO, in regards to iMovie Mac and iMovie iOS you can not sync projects where full functionality is retained.