FAA English proficient is simply ICAO Level 4.
And yes, they should.

Are such English tests being administered to the student pilots? By whom?

By the DPE.

Also by the instructor for the student pilot certificate:
https://www.faa.gov/pilots/become/student_cert/

The english proficient issue was recently brought up by the FSDO, even going as far as saying they would 709 students who came to their attention for english issues, now whether or not they will actually enforce this is a good question. I think a lot of problems would be solved if the students did have better english skills before starting their training.

If the CFI is the one giving the English test, that is a huge bias, and a recipe for disaster. And I am curious how a foreign student pilot "comes to their attention".

Towers report students/pilots who are unable to communicate effectively to the FAA.

I work at Westwind and have several good friends from college that work across the parking lot at AeroGuard. I moved to Phoenix a year ago to finish CFI and work at transpac, but ultimately came to Westwind instead. Feel free to message me if you have questions about th schedules, quality of life, pay, interviews, or pretty much anything about both schools here at deer valley.

brocklee9000,

How many hours do you guys/gals average up at Westwind? Finishing up my MEI on Wednesday and trying to figure out the best place in the valley to instruct.

Thx.

I've only had instrument students since I started working here. I'm averaging 50 flight hours a month. A guy who started with me is doing 60-80 a month with PPL students (the solo stage always makes your hours dip for a bit). I think PPL is where the hours are greatest at westwind. Aeroguard will get you more flight hours with IR students, but you won't start off with IR. My friend over there with PPL students is also getting 50 a month, and my friend who is a program manager and check airman is getting fewer than that. The IR instructors get 80-100 per month. However, the pay is significantly better here. I'm billing 120-150 hours per month at $30/hr as a CFII. I take home in one paycheck what my friend across the street takes home in a month.

Honestly, I'd say work wherever is closest to you. If you're up here on the north side, apply to westwind and aeroguard. If you're on the east side, apply to CAE, UND, etc. Aeroguard also has a location in Chandler. I'm assuming you have other CFI certificates right, since no one is going to hire with MEI only. They all have their advantages and disadvantages. Aeroguard offers insurance benefits, westwind doesn't. Some schools are salaried and set your schedule, others are hourly and let you have free reign over your schedule. At the end of the day, they're part 141 programs and are predominately foreigners, flying in the heat and sharing really busy airspace.

My school has people commuting from opposite sides of the valley, like as far away as Tempe or Gilbert and Chandler, and I honestly can't fathom why. Just as I wouldn't advise commuting or moving for a regional, I wouldn't do the same for a CFI job. The time and fuel costs will really add up over time. Of course, that's assuming you're already here in Phoenix. If you're out of state, obviously disregard that part and just apply to every school. ATP/ASU and UND are at KIWA. CAE is at KFFZ. Aeroguard is at KDVT and KCHD. Westwind is at KDVT. Lufthansa is down at KGYR, they have a nice setup (but require much higher mins). American Flyers is at KSDL, and ATP has a small branch there too. And then pretty much every airport around the valley also has part 61 operations. There is a ton of flying to be had no matter where you go. What it came down to it for me was: location (I live very close to KDVT), the pay (I'm making a lot more than the other offers I got), and how you feel like you'll fit in with the company.
Good luck