drainplumbing
I have the following pipe coming out of the wall. It seems to me that I would need a p trap that screws onto the existing thread but all drain plumbing I have seen so far (e.g., https://www.lowes.com/pd/Keeney-1-1-2-in-Plastic-P-Trap/1076241) appears to have the plastic connecting downward (it has its own thread and you use the top part to secure it to the drain). Am I missing something?
I don't see any problem with installing it horizontally on the run between the two sinks. You'll probably need a new slip nut (the part that goes on top of the trap) but otherwise it can probably connect into the existing union that's there now. Just keep the piece that goes to the RO on the top (facing up).
Much too much up and down! You want a P trap with an additional angled connector toward the wall.
The first down and up from the sink drain is correct. That is the P trap. It should end with a horizontal leg pointing toward the wall (an L shaped piece).
This will not line up with the opening in the wall because the drain is offset. Therefor that leg will run a distance and then will use an angle to bend it back toward the wall.
You could use a right angled slip joint connector (running horizontally).
A 45 degree angle would be better (the less sharp the bend, the less likely to stop up). However it may be hard to fine a 45 degree slip joint elbow. You could use standard pipes and elbows, but these would require glued joints rather than slip joints.
Finally there should be a straight run from the angled connector to the wall connection.
You can rotate the P part of the trap so the L leg coming off the P is parallel to the wall (assuming you are using the 90 degree elbow; if not, adjust accordingly). Then the elbow turns the drain toward the wall. Another horizontal run carries the drain back to the wall.
You need to measure the length of the horizontal leg of the P and the length of the connecting run to the wall; be sure to allow for the sections embedded in the elbow and trim each accordingly. Assemble.
It is easier done than described.
Best Answer
You need to find the rest of that trap or purchase a new one. If you can not find a trap that matches yours you will need to cut the old stuff off and start with a new trap kit. I have found even 30+ year old plumbing matches up to brand new pieces. Take this photo to a plumbing store or big box and they can set you up for 3-15$. The house ghee price if you have to cut and replace the lower for the bend with a nut.