In British English, the standard is 'write to me'. In American English the standard is 'write me'. Similar variants exist with 'out of the window' and 'out the window'. When did the dropping of prepositions start happening? Are there any historical or regional influences? If so, do you know of any research that has been done on these? Finally, which version do other forms of English (eg Australian) use?
Some of the regional dialects in Britain treat prepositions differently. In Yorkshire and Lancashire they will, for example, say 'Give it me' rather than 'give it to me', or 'give it me back'.
In the North East, the Newcastle-upon-Tyne/Sunderland area, the so-called Geordies will say 'I'm asking you for to tell me', thus inserting a 'for' into a sentence where standard English wouldn't.
There are quite a lot of others but I can't think of them late at night.
I was taught that this came about during the early 20th century when there was an influx of German and Russian immigrants into the US. Where German and Russian would have used the dative case on 'me' and thus needed no 'to', English has no similar declensions and so 'Write to me' became contracted to 'Write me'. The form then stuck in American English.
The catch with this story is that I have never seen any authoratative justification for it. Therefore, I can't say if there is any truth in it. It does however sound plausible.
I have never heard the 'Write me' form used in Australia, New Zealand, South Africa or India, although others may have better knowledge. I have heard it used in the Caribbean. Perhaps, due to the proximity to the US there is a greater influence.