You will need to create sharing rules for the objects that you want set like this.
To do this, you will need to either set each of your managers in a different role, or create 2 public groups, each with one manager in it.
After you have done that, on the settings page, search for Sharing Settings. Then, scroll down to see the different objects within your org. Under the object header, click the New button.
Give the Sharing Rule a name.
Set it so that each role or group can only share to it's own role or group.
Set your access level to read/write.
You will need to do this 3 times on each object, one for each manager, and a third that says that if the owner is an internal user, share with owner role.
Based on the comments, you will also need to remove the Read/Modify all flags from the manager. Those 2 flags will override sharing and field lock rules.