If you physically work in New York, you are subject to NY tax as a 'non-resident''
However, you get credit on your New Jersey return for tax paid to New York.
Click this link for more info on How to File a Non-Resident State Return.
However, if you are an 'independent contractor' you may not be subject to NY tax (only NJ).
This link has details about New York Independent Contractors.
The major difference between C2C and 1099 is that with C2C, you don't have to pay self-employment taxes on your income. However, you must pay yourself a set salary, as well as pay all required employee and employer taxes.
Here's more info on Corp to Corp.
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