What would cause sinks to back up into each other? They're on opposite sides of the same wall. Tried Drano. ?

there is a problem the WAY the sinks are connected together.  Maybe they used a T connector. and the block is in the l but not in the - part so you just shuffle water from one side to another.  It should have been a deep Y connector though the blockage could still be in the l not in the V.

So you need someone at one sink and plug up the drain hole and the air break hole (which is that slotted hole) Stuff a wet face cloth in front of it and hold it and cork the drain.  Now block the slotted hole on the other side as best as you can and plunge down with the other hand adding water while you plunge.

I didn't have a plunger, so I used boiling hot water and it melted through the blockage....which is normally soap scum sludge.   That might work for you too.  I did not have Draino or Mr Plumber or a snake.  Never had a problem so won't be buying that junk.

Get another plunger. Have someone hold the plunger over the drain on the other sink. Use the other plunger on the sink on your side.

got 50 bucks for a rechargeable cordless drill?

get a 30-50 ft plumbing snake,a size that can be stuck into the drill opening,,tighten it up,,run a faucet on low pressure,press the trigger,,slow speed,,back n forth in & out,,reverse it,put a plastic bag covering the drain opening to stop goop from splashing up,sounds like the blockage is not too far out,,if.when you get the drain running freely,get enzyme based drain clearer from home depot,it looks like instant tea in color,pour a tbspn into each drain,pour a pan of warm water down the hatch,let sit 24 hrs,

The line is blocked somewhere after the drains join.

You need to physically remove whatever is blocking the drain. A snake is the normal tool.

you're going to have to take a wet dry shop vac and you're going to have to fill the sink up and plunge it up and down and up and down and force that Drano into that other sink wet vac both of the sinks perfectly dry the throw the water outside or down the toilet. then repeat that operation you need to get all of that Drano out of there where eye protection while you're doing this by the way! Then put the stopper in the other sink. Put the wet vac on the open drain not the one that's plugged up with the stopper. And suction whatever you possibly can out with the vacuum cleaner. You might even suck out the clog! When you're done go to the plumbing supply store and get concentrated sulfuric acid drain cleaner. Liquid fire! Get the half gallon container. Pour half of it down 1 sink pour half of it down the other sink. Wait half an hour pour. in half a gallon of water into each sink. Wait another half hour and that clog will be melted and you can run hot water through both sinks!

When you use a plunger, you need to block ALL air outlets, to force water and anything blocking the sink, out to the drain.

Try the plunger again

but first

seal the drain in the other sink (with a tight plug if you have it, with a rag tightly fitted into the drain if you don't have a plug).

If these are lavatory sinks (bathroom sinks)

they probably have an "overflow drain" hole inside the sink near the top

and you will also have to plug those holes in both sinks before attempting to plunge.

A second person holding the plugs tight against the holes in the other bathroom will help.

Make sure you've got a GOOD plunger, one that is not too soft, not worn out. Korky makes a pretty decent plunger for < $10, available at almost any Walmart, Kmart, etc. or hardware store.

Make sure your plunger is well-centered over the drain hole. Water should be at least 1 inch deep in the sink, 2 inches or more is probably better.

The first plunge is going to push air and water out and might be difficult to keep the plunger located over the drain hole. If it "slides", keep the plunger under water and slide it back over the drain hole.

Plunge at least 10 times, 20 times is OK. Try to work the plunger all the way up and down. Do NOT pull the plunger above water level.

If that doesn't work, if it's a bathroom sink, next try a small sink snake. Some are quite cheap.

https://www.thespruce.com/buying-a-drain-snake-271...

If it's a kitchen sink you've probably got grease. You said you tried Drano but that stuff has NEVER worked for me. You can try a snake or a professional-strength drain-cleaner acid that you can find in a hardware or plumbing supply store. ***Make sure to plug all sink holes like you did with the plunger*** before using acid. Make sure to get as much water as possible out of the sink before using acid.

One type that works pretty good is called "Liquid Fire", but anything that you can find that is ***professional strength*** and typically used only by plumbers is probably going to be just as good.

SO, recap

- try plunger again, making sure to seal drain holes except for the one being plunged

- if that doesn't work, you can try a snake or professional drain cleaning acid

-- I work for Plumb Crazy Plumbing in West Memphis Arkansas

They share a common drain and the common drain is plugged. Get someone else to plug up one drain with an orange, a ball or a balled rag, then plunge the other sink. Messy but effective.

If you want try a plunger, you'll need to block one sink. There is apparently something in the pipes after the two connect.

The cause is a clog below the 2 sinks.