When using a water-pressure machine to clean a house pre-painting, does the water pressure at the house matter?

There's certainly a minimum volume of water that needs to be fed into the machine-- they'll work with surprisingly low pressure as long as the volume can be fed.

I don't have my fluid power book with me, so I can't pull out the math, but the pressure-washers work much like an electrical transformer-- they convert "low voltage, high amperage" (e.g., low pressure, high volume) water into "high voltage, low amperage" (high pressure, low volume). So it should be able to take any pressure of water, I've seen gravity-fed truck mounted versions on the streets of SF, but there's likely a minimum volume (gals/min) that your water system would need to provide to keep up.

My painters are not using a professional water-pressure machine...

there's the problem. you're house is not getting properly hosed and you're getting hosed in the process.

so to speak.