(My answer to this question at Quora.com)
There are two issues here: speaking in a reasonably native accent in more than one language, and speaking with more than one native accent in the same language (e.g., various regional accents in England, Scotland, Ireland, the U.S., Canada, etc. when speaking English).
It is common courtesy to try and adopt the standard pronunciation of a country when you learn its language—if only so as to make yourself more intelligible to the locals and to put them more at ease. But switching between different native accents in a given language requires training, and is something that you would do only if you were a professional actor or standup comedian and needed it for your work.