I might correct people but I'd rather not waste my time as they're bound to do it again....and again...the exception being businesses or finance where it does matter.
Someone once wrote my name down as Rita. Now, I wouldn't mind the name if it was mine but it isn't. Not even my middle name.
My moniker (and my real name) Gabriella is sometimes misspelled on here but we can all get a bit careless at times however, I always spell out my surname after saying it and the numbers of people who DON'T pay attention and still send out mail with it spelled the way (in their tiny minds) that they think it should be spelled, are legion. 
When I ring to correct them, the response is nearly always 'Oh, I thought...' and I say, 'Perhaps you didn't think. I actually spelled it out for you yet you spelled it the way you wanted to.'
Yes, I really do. They are idiots if they don't listen.