The simple answer is that the cards should be accurate. Whatever the person's real position is, the measure of a good business card is one that's correct. Of course, that's the straightforward reply.
Now let's get to the more complicated part. Business cards are an indication of what a person's responsibilities are at a company. A business card is the first impression you make on your customer. According to Eric Rumbaugh, attorney and employment law columnist for Contingent Workforce Strategies magazine and CWS 30, there are two different problems presented by the issue of business cards. "The first and biggest issue here is basically one of liability. If a business sends somebody out in the world to do something, the business or company potentially can be held liable for the acts of that person as its 'agent;' and the second issue has to do with co-employment" says Rumbaugh.
Now it's time for a few examples to make it clearer. If company X sends someone to drive to Denver and that person gets in a traffic accident, company X can be responsible for the acts of its 'agent,' regardless of how the business card reads. Likewise, let's take the example of insurance companies, which often use brokers on staff to sell insurance, but may also engage independent contractors to perform the same function. Both are considered the insurance company's "agent," and as such, the company has potential liability for their actions in the course of dealings with its customers. So if either one, the staff broker or independent contractor, does something fraudulent in the course of dealings with the company's customers, the company has potential liability for his or her acts. And at that point it does not really matter what the business card reads. The insurance company is still potentially liable.
But a company can be liable for the acts of an "agent" (and employee, contractor, or anyone else acting on its behalf) beyond the agent's actual authority, because the agent has what's called "apparent authority." If company Y hires someone, either as employee or independent contractor, to buy 10,000 pounds of coal, and that person also buys 10,000 pounds of wood, company Y can be responsible for paying for the wood even if it didn't want the employee to buy it. The "agent" may not even have had the authority to buy wood. But if his business card says "buying agent," it looks to the seller that the agent does have the authority to buy wood. So the seller ships both the 10,000 pounds of coal and the 10,000 pounds of wood. Even though the agent was not authorized to buy wood, company Y might end up owing for the wood. This is because the company has armed him with what the law calls apparent authority. The business card goes a long way toward creating apparent authority.
Make sure when your company prints a business card for somebody, the title printed on it reflects what that person actually does. "That is going to be part of determining what their apparent authority is. You don't want to inadvertently create apparent authority for someone that exceeds their actual authority," says Rumbaugh. The biggest source of liability is having a business card that's inaccurate, that's misleading, and creates apparent authority for someone that could create liability for your company.
So in answer your original question, in most cases, your company is a joint employer along with the staffing firm that leases its employees to help run your MSP operations. The business card can just state "[MSP] liaison" and does not need to mention the staffing firm's name, as the contingent worker is in your company's building daily working under your supervision. Only if the contingent is an independent contractor does the scenario change. Then the business card should say "independent contractor affiliated with [your company]." If there is an issue over misclassification, having a contractor with a business card with language similar to that of an employee would help build the case against your company.