Power Steering help!

So the pump in my 2000 328i failed. (The shaft apparently broke inside) With 175K it didn't really surprise me. So I ordered this "new" one I found on eBay and gave to my mechanic to swap out. The new one did not come with a pulley so he used the one off the failed pump. Every thing worked fine and I drove it for 200+ miles that night. There was some shuddering but he said that there was probably still air in the system and that should work out in 1 or 2 days of driving.

The next day I drive it to and from work (30 mi each way.) When I get back the pump is at that point screeching intermittently. Upon further inspection I found the bolts for the pulley are loose so I tighten them. The pump still looks like it’s new with no cracks or leaks. Now the pump is screeching almost constantly. When I turn the wheel while the car is stationary the pulley moves almost an inch toward the radiator and screeches. Has the shaft broken again or is the pulley slipping off the flange?
So do I need another new pump now or just the pulley???

If the whole pump is moving you need to check the mounting bolts. It's not too difficult from underneath the car (even from the top is not impossible w/o removing anything but the belt), and you may not even have to lift it up to do it (though it would help). Remove the belt so that there is no tension on the pump, and check every bolt/screw on it. Find a DIY if necessary, some pumps have three mounting bolts and others two.

The mechanic sounds incompetent; maybe he's new. But air in the system is not something that works itself out over "1 or 2 days of driving". It is a hydraulic system and any air is always problematic.

It doesn't seem to be the whole pump moving just the front portion when I turn the wheels and gas it a bit.

Remove the belt and check the pulley and pump shaft.

Did you tighten the pulley with the belt on? It sounds like the pulley is still loose, if it is moving but the pump is not. You'll have to remove the belt tension before tightening the pulley bolts. And if it was never attached and tightened properly, it might not be mated to the pump properly. There are manufactured concentric circles that must align upon a single plane.

So finally got the time to look at it again. Took the belt off and the pulley came right off the spindle with the flange. So apparently the pulley and flange were just spinning on the spindle. Looks like i need another pump now.