They don't need to be paid. The inbox is where you receive posts from other people. Public inbox is for replies, zaps, likes, etc to your posts. Private Inbox is for your DMs.
You want everybody to be able to download replies to your message, so your Public relay choice should allow that.
You don't want other users downloading your DMs, so your relay choice for the private inbox should also support that.
But you can put public relays in the Private inbox if you don't mind other people seeing the encrypted DMs going to you.