It only leaks when the hose is connected to the sillcock and under pressure and the leak is coming out of the stem area. The sillcock is a Matco-Notca model 1019A.
Agree with the previous posters regarding legal stuff. Dot the i's and cross the t's before proceeding. Get it in writing. HOAs are famously litigious environments and the legal fees could easily exceed the plumbing fees if things go wrong.
Are you sure about that model number? Looked for it and only found some shut off valves which also had 1019 in the model number. Can you see how the sillcock is attached to a pipe coming out of the wall, or is this one of those freeze proof models which itself extends into the wall? Posting a picture would help, with the location of the leak marked.
Anyway, if I understand what you wrote, you are saying that water comes out "the stem area" when the spigot is open and something is attached on the downstream side, like a hose with a closed nozzle. Starting at the handle and working in you will (usually) see a screw that holds the handle onto the stem, the handle itself, the stem (usually, there might not be enough clearance to visualize it between the handle and the next component), a nut, and then the body of the sillcock. Is the water coming out at the junction of that nut and the stem, that is, from the little "O" shaped gap between them? That happens when the packing in the nut around the stem fails. To fix it one generally: shuts off water to the spigot (often to whole building), removes the screw, pulls the handle off, removes the nut, scrapes out the old packing, puts new packing in (often made of teflon with a diameter maybe twice as large as a piece of spaghetti), tightens the nut down to compress the packing around the stem and the top of the valve body, puts the handle back on, puts the screw back in, turn the water back on. While in theory this is a simple job things can go wrong at every one of the those steps. Of these the most catastrophic would be snapping an old pipe in the wall when using a lot of force to unscrew the nut, which may very well not have been off since it was installed when the building was constructed. I think most people would suggest also replacing the washer on the end of the stem at the same time, since it is already apart.
Have you tried other plumbers? How about a handyman? Their business is more about fixing things, whereas for many plumbers it is more about replacing things. Both should have the tools and expertise to repair this valve.