"I want to come there" or "I want to go there"

I have always thought that go has a slight connotational leaning to describing the journey, or leaving the origin. Come has a connotation to arriving, or the destination.

Thus, "I want to go to Paris" states that I want to make a journey, with Paris as the destination, but that the journey is important.

"I want to come to Rome" implies that the arrival in Rome is for a specific reason that is personally, emotionally important to the speaker.

Moreover, coming to Rome, as opposed to going to Paris, seems to imply that the speaker is addressing someone who is in Rome, and will be coming to them or going to Paris.

I believe that both choices are grammatically valid, but that each has a slightly different meaning invested in it.

Generally, when speaking with a person that is away from you, I want to come there would be understood as I want to travel and arrive where you are.

I want to go there could mean I want to go in the place we are talking of.

The most common/natural expression here is probably along the lines of:

I wish I were there.

(Okay, so many people don't bother to use the subjunctive - as you should - and say was instead of were.)

A possible alternative is I want to be there - it's slightly less natural sounding to me though, at least in British English.

I want to go there. might be an appropriate response if someone mentioned a foreign country/holiday destination that sounded appealing to you. I can't think of any real-word situation in which I want to come there. would sound best.