Hello and thank you for your question.
1099-Misc is generally filled out on a cash basis (after amounts have been paid).
Your taxable income is what it is, however, regardless of any 1099-Misc issued for you.
ie... if you are a cash basis tax payer and properly report all of your revenue on a cash basis, then you will be in accordance with the tax law (assuming you are, per the tax law, allowed to be a tax basis cash payer and follow the applicable rules of course).
Example: You do a job for me between Dec 2010 and January 2011. I don't pay you until the job is done and you are a cash basis tax payer for purposes of this contract. I choose to 1099 you in 2010, regardless (I would probably be wrong to do so, yes). You report your income as you should, in this case, regardless too.
Keep documentation and note when you have received the cash.
You cannot have constructive receipt of the cash (ie... if you received a check before Dec 31 and just choose not to deposit until January, you still have a December receipt).
If you save your documentation (ie... meet the Internal Revenue Code substantiation requirements) and you are reporting revenue properly in accordance with your method, then you will be correct and prevail in the event of an IRS audit / notice.
You should also consider getting a corrected 1099-Misc filed as applicable by working this out with your contractor.
Perhaps, the person hiring you is trying to push expenses into the 2010 year that are really for 2011...
I trust this all makes sense.
Please do not hesitate to ask for more clarification. Thank you again for your question!