Private vs shared teams and how to make them appear in Jira issues

[Miguel Rodriguez]

I am a project administrator in a Jira 9.4 Data center instance. There are many projects in this instance, and many of the projects are managed by individual Business Units, each one having multiple teams and the instance managed by a separate group of instance admins. I found that to make the plan teams to show in the issue field "team" I have to make them shared, which eventually will have hundreds of teams our teams do not work with. 
Is there a way to have only my project teams appear in the teams field. It looks to me that a private team exists only for the plan but does not update Jira issues.