It depends. The most likely scenario is that you have to fill out the Allocation screen on the New York nonresident return. If someone works for a NY based company as a nonresident, NY typically requires the company to report all of the income to NY and then require the taxpayer to sort it out on the tax return through allocation, which tells NY how much of the income was taxable to NY. There is a reason for this, as I describe below.
To allocate, there are a number of methods you can use. If you know the exact amount you were paid while training in New York, then use that amount. If you are unsure, you can figure out the percentage of time you were in New York versus the total amount of time your worked at that job in the year, and multiply the percentage by the income. (It should be close to 50% given the information provided).
There is one exception to this, and it's the reason why the NY business does not divide the income for you. If you are working for the NY company as a telecommuter (working from your home). NY considers you to be working in NY using a provision of their law called "convenience of the employer". If this is your situation, then NY will argue (and enforce) that all of your income is taxable in New York.
The following link has a PDF file of the form where the allocation is reported, which may help you to understand how it is calculated:
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