Can I share my syncs with teammates or can they choose items to sync?

There are a couple of solutions to empower your teammates to control what syncs, depending on how much you want to remain the "gatekeeper", if you will.

Solution A

You can create a filtered sync between two boards which will effectively let you and your teammates pick and choose which cards to sync. The filter will be the trigger for syncing a card. For example, you have a sync between Project A and Project B, you could use an arbitrary label named "Sync to B" and filter for any card that has that label. Once you create this filtered sync on the board, this would enable your teammates to effectively select which cards to sync to Project B by adding the "Sync to B" label to a card. 

Here is a short video on setting up a filter:

Downside to this solution: You need to create and manage the syncs for your team since they cannot access the settings

Upside: All teammates that have access to your project are able to choose any card to sync to any of the team boards you setup, without them needing any access to Board Genius.

Solution B

You can share you Board Genius plan with your teammates. They are then able to create their own syncs - either by selecting individual cards to sync or like the filtered sync above.

Downside: This would most likely use more sync bots, requiring a higher tiered plan. Why? Because User A may select 10 cards to sync from Project A and Project B - that would count as 1 sync bot. Then User B may select 10 other cards to sync from Project A and to Project B - that is another sync bot because they cannot share the settings with User A. So now there are 2 sync bots operating on the same boards.

Upside: Each user is flexible to create their own syncs.