Consultancy firms often have an offshore team in a lower cost country (India, Phillipines, Mexico). The consulting firm typically employs a few people in the US on H-1B visas, contracted to the client, for communication with the offshore team, onsite support for the client, and generally managing things.
But the client often has their own employees on H-1B visas as well. This is unrelated to either offshoring/consulting.
Eg. Take a company like Apple. Apple hires employees on H-1B visas and pays them well.
But also Apple uses consulting firms (both US-based and offshored) for specific projects/functions. These workers are not Apple employees. They are employees of the consulting company -- or in some cases, there is a chain of contracts where they are employed by one company, which contracts them out to a consulting firm, which then contracts them out to end client (Apple).