I am not a tax professional and what I'm sharing is just my experience as a long-time independent contractor. We are considered independent contractors in the eyes of the IRS for our pet sitting business. Schedule C and 1099 are two completely different forms. Schedule C is the tax form you file with your income taxes that reports your income and expenses for your business. 1099 is what a business may issue an independent contractor when they pay them over $600. It simply reports to the IRS how much they paid the contractor.
There is an IRS publication that details how to fill out the Schedule C, and I strongly suggest you get it as soon as possible. There are many expenses you can claim for your business - mileage to and from clients, pet food and supplies, classes, just to name a few.
Rover will not issue a 1099, and Paypal only issues them at a $20,000 threshold (I think). Your best bet is to keep track of your Rover (and DV) earnings yourself. You don't need any forms to back up the income like the W-2 if you're an employee.
Also, if you plan on making a lot of money with pet sitting, it's a good idea to start paying estimated taxes quarterly. At least set aside about 20-30% of your income for use when filing your 1040 in April. You will be charged self-employment tax when you file your 1040, and that often comes as an unexpected and unpleasant surprise.