Whether your business is an escort service, companionship service, or an entertainment company there is no reason to ever hire a driver to drive the independent contractors to calls, houses, or events. Your independent contractors must drive themselves or have their own driver to transport them. Your business should never have interactions with their drivers, if they have one.
Hiring a driver for other reasons is usually a good idea though. That driver should work for your business and not for the independent contractors. The driver's job description would be to collect your fees from the independent contractors as directed at the time and place that you instruct. Really your driver is on-call for you during pre-determined hours.
An added use for a driver would be as a chaperone at a party if you are booking bachelor parties or other celebrations and providing strippers or dancers to entertain more than one person. A dancer or stripper will often have her or his own chaperone – usually a friend or boyfriend, but it's a good idea to have your chaperone present for all party-type events, and collecting fees at the event is convenient when you do it this way.
Paying your driver can be done anyway that you want to do it, but here is an idea that's fair to all involved:
To have your driver on-call for your business from 8pm to 6am (or whatever hours) you will need to pay that driver a specified amount per shift, whether you call the driver or not. All of this depends on how busy you are, and when you first open it's a better idea to be your own driver. This type of pay should be in the form of a weekly pay check, but your driver could also be an independent contractor and receive pay for each shift. Read IRS guidelines on independent contractors v. employees and decide which best describes your situation.
If using a driver as a chaperone for parties or events that you dispatch dancers or strippers to, that driver should receive added compensation. The driver would already be receiving shift pay from your business if attending a party, but this is an extra duty and that shift pay is really only to collect your fees. It is a good idea to have a surcharge – the amount is up to you, but don't cheat your driver/chaperone. A surcharge of $20 per dancer/stripper added to the total tab shouldn't be an issue for the customer. You can explain it just like that – we have a $20 surcharge per dancer for chaperone services, and sorry, but we require that a company chaperone be present at all events. Your driver/chaperone should receive the entire amount.
Other compensation for the driver that is a chaperone at events should include a tip. The dancers/strippers should value the chaperone services enough to part with a decent tip as they are tipped by partygoers. In America services are rewarded with tips. Sometimes dancers/strippers try to be cheap though, so there should be a set percentage. You'll never be able to keep everyone honest, but then the chaperone is there and will see whether tips are flowing or not. If the driver feels cheated on the tip – he or she must deal with you, and not with the individual stripper. You will deal with that stripper. The percentage would vary from 10% to 20%, depending on how many partygoers and strippers are at the scheduled event. You'll have to offer a solution that is acceptable to all – the chaperone and the strippers.
If you don't already understand this I will state it once more: Your company driver should never transport an escort, entertainer, dancer, stripper, or companion to a scheduled call or event. Their transportation must be their issue. Your driver should also never appear partial to a particular person as he or she is there representing the company and shouldn't accept any tips for picking up your fees as you direct. All involved should be clear that your company driver is exactly that, and not their ride or tool to more calls or events.
Edited to add: Your driver should always be reimbursed for expenses. Expenses are gas, tolls, or any other expense that is a result of carrying out your instructions.