Hi, the ChessUp 2 (online board with Embedded AI) does have display rotation: one button press (the screen rotation is not auto-detected) and you're good to go ! Plus, in the ChessUp 2 all 64 squares have LEDs showing moves to both players simultaneously, so even without looking at the screen, you can see suggested moves. The Chessnut EVO is a different "beast" entirely: the big screen (12.3") appears "fixed" on one side, but images show the screen can tilt/lift up at an angle for better viewing. However, LED squares still work for both players regardless of screen position, so they still help. Other online boards (Millennium, Tabutronic, etc.) connect to your smartphone/tablet via BT/BLE, and use your phone as a "second display", just place it on your side when playing Black and you're good ! Ciao 
Electronic chessboard where the black side has a move display screen in front of it?
Yes, but what the old electronic chess boards do is reverse the board - usually an option (often vague) although some will switch automatically i.e. at the start of the game you force the computer to move first and the computer will move from the top.
Some low level (beginner) chess boards unfortunately expect you only to play White - which is quite annoying as you have found. This limitation is never mentioned in the packaging or the manuals! Some will not even mention you can play Black from the bottom of the broad and you have to figure it out by trial and error.
The only old chess computer I know off that has dual White & Black displays is the excellent Excalibur Grandmaster - of intermediate strength but full OTB size. Unfortunately the cost of this old machine is on a par with much stronger modern machines like the Millennium Chess Genius Pro etc, so is really only a collectors item and hence the cost.
Well, unfortunately it seems like dual displays or "auto-rotation" remain rare features even in modern chess computers. Most manufacturers expect you physically rotate the whole board, that is annoying, use a connected smartphone/tablet as your second clock, or just play from White's perspective only. This is surprising given that the Excalibur solved this problem decades ago ! It's clearly a "cost-saving" measure that persists across the industry. The Millennium Chess Genius Pro does not have auto-rotate display, but, as a "workaround", users report setting Black pieces above the display and "ignoring the misleading coordinates". Both the Chess Genius Pro and the Tabutronic Proelium (by italian Certabo) can connect to external devices (phone/tablet) via Bluetooth, and using THAT device's screen as a "secondary display option". On the other hand, the ChessUp 2 (which is an online board with "embedded AI"), as said yet, has manual screen rotation (one button press, not auto-detected). Ciao !
With the DGT Centaur, you can play without rotating the board. Admittedly, it isn’t one of the cheapest chessboards, and it can’t connect to the internet (unless you modify it).
I should have mentioned that the dual-displays of the Excalibur Grandmaster are actually intended for TWO human players to use - ie. both have access to the computer info. I do wonder if each human player could configure the engine differently - if so it's not a bad idea as it would have some unique use-cases for such an old machine.
FWIW: The Millennium Chess Genius - even the older designs as the engine and opening books are the same - are by far the best machines in terms of strength/features to cost.
I am obscenely lazy, and don't like the old electronic chess boards that only have a single display screen in front of White, telling you how to move the pieces for the computer's move. It's a pain to have to rotate it after each move if you're playing Black.
Are there any chessboards that either have two screens to show requested moves (one for each player), or one in front of the Black side (since I already have one with White)?
Help a lazy guy out.