After discussing with svk in telegram, I know now how it works.
Issues.
1) You should not be able to add your own public key as contact, it shows 'missing' in your private account then
2) The receipt only shows in the console, but it should be in the dialogue
3) Entering the receipt works, but it should display more details like how much has been received
4) The balance is only shown in the transfer page, and only there if you do a refresh of the page (and unlock)
5) I would expect to see ALL my balances in the dashboard, including my private balances.
6) There is no link to send txs to private contacts, but there is a link for private accounts to send from. Inconsistent.
EDIT:
7) I created a new private account and tried to transfer funds from my public account to ~test3 (my private account). Got a receipt in the console which I tried to enter in 'Receive'. Getting this error in the console: Error: Missing required field: Label is required unless a private key is provided(…)