You have complete control over what your partner can and cannot see.
Transactions: To edit whether your partner can see a specific transaction, tap the avatars next to any transaction. As long as you added the account, you'll see the ability to mark that transaction as Private. Doing so will prevent your partner from seeing that transaction.
Accounts: Tweaking the Shared status of an account is easy! Just head to Settings > Accounts. From there, choose the relevant institution. Next to the name of each account, you'll see the current status (e.g., Shared, Visible, Just Me, Ignore). Tap that and you'll be able to edit it.