A solutions/services company doesn't run differently per se because they use open source software in their offerings.  The open source software provides additional tools to be used in the vendor-client relationship.  If the OSS project is run by the vendor itself, then marketing has new tools in terms of getting the word out more widely about the value of the solution (providing you don't confuse community and customers).  Even sales gets the boost of a [hopefully] more trustworthy public image based on community engagement practices.  

Because the open source software provides additional tools, a vendor may choose to make more use of these new tools, and hire accordingly, but it doesn't change the roles of sales or marketing.   

I'd like to add one point to the discussion: System integrators who built customer-specific solutions based on FOSS will sooner or later think about tweaking these solutions to become more generic and to bundle them to offer an industry-specific out-of-the-box solution.

This is a tempting strategic move and FOSS makes it more likely to succeed and switch from being a system integrator to actually become sort of a software vendor. Reason being that FOSS compenents conveniently allow system integrators to add value on a higher level of the stack.

Nevertheless, such a move should be considered carefully, because the change of business structures and processes as well as the corporate culture is tremendous when transitioning from a system integrator to a vendor business.

Having said that, perhaps with FOSS, the distinction between system integrator and vendor is sometimes blurry to say the least.