Currently, there is only one way to access a private trade that someone else sends to you, which is within the notification automatically sent when the trade is created. (Or if they send you the link, I guess that counts.)

With how easy it is to delete every notification in one's inbox, there needs to be a different place to access them. There isn't even a confirmation button when deleting all notifications, so it can literally happen accidentally in the blink of an eye due to a simple misclick.

Let's say, hypothetically, someone has set up a forum post buying Crunchy Worms at 20 GB per, requesting that others send private trades for them to buy out. But at some point, they scroll down in their notifications and manage to delete all of them. There's now a banner at the top of their screen saying they have a new message, but there isn't anything there anymore because they've been somehow unlucky enough to delete them at the perfect inopportune time to miss whatever the new notification was. If it did end up being a private trade sent to them, they basically have to rely on that player messaging them after the trade expires, wondering why they didn't accept. Since they never saw the notification, they can't exactly initiate contact on their own.

That example is probably a tiny bit extreme, lol. But not impossible. Even just finding the damn notification might be difficult if you have a ton of branch item sales in the same period of time.

A good, simple thing to add would be either a tab within "Manage Trades" similar to the separate tabs for "Hidden Trades", or a section within "Your Trades & Offers" below "Your Trades". (My, that's a lot of quotation marks!)

If it were its own tab, I'd probably label it "Private Trades" and move everything pertaining to private trades there, which would have two sections - Outgoing private trades and Incoming private trades. Otherwise, just a section for Incoming private trades would be sufficient.

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My other suggestions:
Masterpost here