In many jurisdictions, the US included, whether or not you are a contractor is not up to you and the company you do the work for. E.g. even if both parties think it’s a contracting relationship, the IRS can decide otherwise.
The nature of the relationship, and the work you are doing determines whether your employer is a regular employer and what taxes they have to pay. In Canada, for example, certain taxes (GST) are applied to the contractor relationship based on whether the product the company sells is available within the country, and the distinction between contractor and employee is much less fuzzy than it is in the US (if you have to ask, the person is almost certainly an employee).
The nature of the relationship, and the work you are doing determines whether your employer is a regular employer and what taxes they have to pay. In Canada, for example, certain taxes (GST) are applied to the contractor relationship based on whether the product the company sells is available within the country, and the distinction between contractor and employee is much less fuzzy than it is in the US (if you have to ask, the person is almost certainly an employee).
It gets complicated quick