I think the ultimate goal for all of us non-native speaker is to be fluent and to sound just like a native speaker when we are talking.
However, most of the time our accent comes out unwanted: the way we pronounce a certain sound, the pitch, the intonation, some pauses may be influenced by our mother tongue. Or sometimes our pronunciation is just too academic.
I am happy when someone makes a compliment about my good pronunciation but I also really like when I can guess the cultural background of a person just listening to his/her way to speak. What do you think? Is it more important to sound perfect in a language or to keep some dearth as mark of who we are?
I agree with what Olga & Ewa said.
We should do our best to pronounce words like natives, in order to be clear and understandable. (This is our goal)
Not to hide our own identity