What's wrong with this car?

My daughter just got a 2008 Ford Escape. She called me panicking because she said smoke was coming from her tire. It's the brakes, but she thought it was a tire. She took it back to the shop and they told her it was okay to drive to work the next day just to keep an eye on it. Of course they told her that. They got the car sold so they don't care. So she has been driving it but I'm worried. We both work during the day so sharing a car would be rough and she works out of town. Itll be a week before she has the money to fix it. Should I let her drive it? It has not done anything since the day she called me crying

I wouldnt let her drive it, either drive her or get her another ride

I wouldnt want my daughter driving that car. Brakes could go at anytime

Don't let her drive it. Get it towed to a mechanic to check the brakes. Check the coolant. Escapes of this vintage are known to blow head gaskets.

I had a similar issue where I could feell the brakes grabbing while driving and they began to get hot and smoke as well. I took it to the shop and they told me there wasn't anything wrong, which clearly they were wrong. I drove it like that for a month or so until I took it back to them again and demanded that they figure out what was wrong. The problem was the hose going from the master cylinder leading to the front brakes. It was failing. They replaced that and everything has been fine. The other thing it could be is the brake calipers, they may need to either be cleaned or replaced. I would say it was safe to drive, just do not let the brakes get too hot to the point that the disc is glowing red. That happened before in another car I had.

The first step is to take it to a mechanic who can tell you what is wrong with it. It's not coming from somewhere inside the engine, so it shouldn't be really hard to figure it out.

Brakes are there to stop the vehicle, so if in doubt don't drive it.
It COULD just be where the car has stood for a while at the dealers and built up some rust on the rotors, or the parking/E brake may not have been fully released.
Check the brake fluid level under the hood and take it for a short test drive yourself, you should be able to feel if a brake is dragging or maybe replicate the fault.

It's just the brakes overheating because of a sticky brake piston. It's fine for short journeys. Quick and cheap job for any shop to fix.

If it was sticking brakes, she would not have driven anywhere… As the brakes would be locked on. So a wheel is not turning AT ALL. Front, rear, you did not say. From the exhaust pipe or what; when the car is first started in the morning or in the middle of a drive.
The missing information is so important.

Whatever it was; it stopped. So not even a mechanic could figure out what is wrong because the problem no longer exists. Ceramic brakes don't smoke. If there was any oil on the rotor(like if it was a new rotor) then that burns off and is gone. So, nothing to see. 2008 is not that old.
. The shop was irresponsible by saying "keep an eye on it" For the mechanically illiterate, keep an eye on WHAT EXACTLY? I should not need to do THAT. The shop HAS TO CARE, FOR IT IS THEIR RESPONSIBILITY and they can get sued or even SHUT DOWN PERMANENTLY. Which is way bigger coin than making sure the car is good.
Have her explain to the mechanic the color of the smoke and how much and the smell of the smoke.( a description relating it to something else helps) If it was a coolant leak then it would still be steaming a week later. Rotor smoke, or brake smoke is so small I never notice… Either visually or smell it.
If it happened when she started the car from cold… And the smoke came out of the tailpipe, then the smoke is just steam as moisture inside the tailpipe STEAMS OFF. That happens with every cold car. In the winter it happens all the time. THAT IS NORMAL. Steam has no smell as it is just water vapor coming out of the inside of the car exhaust pipe and muffler. It is also related to how humid the air is. During a rain, the air is more humid than during a long dry spell. More humid means more moisture inside the pipe after 4 hours of cooling so, more steaming. All cars do it.
So, could be totally NOTHING. It never showed up again means it is gone. No worries.

40+ year mechanic.

"She called me panicking…"

So, you're basically admitting that you did not properly prepare your daughter to handle the little problems in life, and that you also failed to teach her common sense things like having a proper pre-buy inspection done by someone qualified to evaluate the condition of a used car.

EPIC FAIL on your part.

The problem is overheated brakes, either from riding them while driving or from one of the brakes not releasing properly and dragging against the brake disc or drum, or from driving with the parking brake on. Until the cause is determined, I suggest that if you don't have far to drive to work yourself that you switch vehicles until it gets evaluated and repaired.

It sounds like a stuck caliper piston or a bad brake hose. Try stomping on the brake really hard, as if an emergency stop. This may free up the piston. This was taught to me by a dealer. It needs to be jacked up when hot to see if the wheel will turn by hand. Something is wrong but I don't know if it is safe to drive at all.