How much should I reserve from my paycheck when I need to fill out a 1099?

Simple answer: 23%.  A more complicated rule of thumb: 14% of the first $12,000 and  25% of the rest.

But taxes aren't simple. It depends on your total income and filing status,  for the year. 23% will be adequate for up to $50,000 of total income.  If you expect to make less than $12,000, then 14% is enough.

This assumes that you are working as a self employed person / independent contractor, and not a regular W-2 employee.  You do not "fill out" a 1099.  The companies your work for will send you a form 1099-Misc.

As self employed person, you have to pay your own social security and medicare ("FICA")*.  This is about 14% of all your net pay.  This is in addition to income tax of 10-12% on amounts over $12,000.

*Note: The term "Self Employment Tax" (SET) is used for how the self employed pay their social security and medicare tax.  SET is paid thru the normal filing on an income tax return, along with any income tax due.