How do I find a lost Euro mobile phone in the USA? I do have GPS but not roaming. I have searched bins and restrooms and reported it to the police. I cannot phone my server.
Both of these services require you to have opted in the device to the corresponding service before losing it. If you don't remember whether you opted in, you can always try to find the phone using the service.
You are most likely out of luck.
That the phone has GPS on means that it knows it own location, but since roaming is off it cannot transmit that location back to you or anyone else1, even if it is configured to do so with services like the ones describe in Michael Hampton's answer.
If the cell phone were connected to a USA network then the carrier would know to which tower it is connected to, but:
In a city that would give you a search area with a radius of a couple hundred meters at best (and a much larger area if it is in a rural area).
Worse yet, carrier companies do not give this information to particulars (too much risk of giving away private information of customers) and if you tried to follow a formal procedure (convincing a judge to write an order for the carrier to provide that information2) by the time you could get the location the cell phone battery would have died.
Also, the absence of roaming means that most likely3 your phone is not connected to any network anyway.
In short, the fact that the object is a phone with GPS does not give you any advantage for searching it, so you must rely in the same methods that you would use to find any other object of similar size, like a wallet: contacting local lost&found services, remembering when was the last time you used it, remembering all that you did after that, even following again your path that day...
1 Unless it is configured to use some WiFi connection that you have not told us about.
2 If that were even possible at all.
3 Even without roaming you can make emergency calls, but I do not know if that means that the phone is actually always connected to a network or if it connects to one only when it has to do the emergency call.