A construction design or architectural design assistant is not a licensed architect - much like a paralegal, they do the work of helping turn the architect's vision into reality.
Because of this, they are normally hired by architects - if you are hiring a team to design a building, then you will be hiring the architect and their assistants as a package. Architects hire design assistants to do computer aided design work, draw up plans and work as a liaison with the client. They may also inspect the project during construction, do materials research and design minor alterations and additions. They create the schematic and working drawings construction managers need to complete the project.
In other words, a construction design assistant is a person who does the ordinary, tedious work of design so the architect can focus on more complicated matters. The only people who would normally hire one are architects, although landscape design assistants are sometimes employed by local governments and park authorities to handle routine seasonal work. Facilities managers may also hire one for remodeling projects for which an architect is not seen as necessary. Some also work on construction sites for the manager.
However, people may not need to hire a construction design assistant directly, but anyone who has, say, had a house built, has had one working for them at some point - often as the person from the architecture firm they actually talk to. Construction design and architectural assistants tend to come into the jobs right after college, sometimes with a Masters, and may or may not work up to being a licensed architect themselves. Some of them may also be interns or summer employees who are studying architecture.
Putting together an architectural design is a complicated task that requires multiple people to complete successfully.