You must be a subcontractor. You received the money and the IRS knows it. If I understand what you are saying, there's something shady going on or those checks were supposed to go to you. They have your social security number. So it has to be income and those guys are writing what they "pay out" as an expense on their own taxes. Go to those people and tell them you want a W-2. It could be that you are not an independent contractor. It depends upon IRS definitions, e.g, you have no control over what work you are to do; you have to perform when your employer tells you to and you have no control over how you are to perform the work.
Make several copies of the letter your employer wrote. Keep in several different safe places. This one thing may save you a lot of money and hassle down the road. In the meantime, save all your receipts and keep an accurate, detailed log of all of your expenses. This will help a little as you have extra tax to pay. You have to pay a 15.3% self-employment tax unless your gross income is under 400.00. You get a credit for half of that on the 1040. If you don't know the mileage you drove, look at the State inspection receipts. The mileage is there and you can interpolate. Think of work related cell phone calls, look at your bills, etc. If you are sent a 1099-MISC with Box 7 filled in, you have to pay what the IRS calls Self-Employment tax. You have to fill out a Schedule SE-Form 1040 and eFile or mail along with the other income tax forms. This is actually social security tax and none of us can get out of that tax either, unless you register as a member of a "religious order", as the IRS calls it. If you live in one of the 43 States that have income tax, you may have to pay State income tax, also. I know all this because I had clients in this very same situation. Don't give up. Get something to hold your receipts: a shoebox or anything to start. As the years go by, you will become better and better organized.
Complete an IRS Form SS-8 to get an official ruling on your status. This will help you get unemployment if you get fired. When you file your income tax return, you can attach Form 8919 Uncollected Social Security and Medicare Tax on Wages and only pay the employee's half of social security. You will still have to cough up all the income tax. IRS and the states are stepping up enforcement as there is severe abuse. These people are trying to get out of paying Workers Compensation premiums and their portion of the FICA tax. An employer has to insure employees against on the job injury by means of Workers Compensation insurance in every single state if he has more than two employees in most States. The IRS is aware of this severe and growing problem and began cracking down on it hard about 3 to 4 years ago.