The short answer: ANY ABLE BODIED HUMAN BEING.
Like real estate agents, who offer a great deal of service to identify, price, market and sell houses but aren't legally necessary, architects do not need to be commissioned to design a single family dwelling. No, no one licensed is required by law to design a house. If you would rather do it yourself, it is completely legal.
If you can design a house and create the minimum drawings required by the local building department (with structural engineering and any other consultants required in your area) to qualify for a building permit, you do not have to pay anyone else to do it. You can also build the house yourself, without a license in nearly all jurisdictions, as long as you meet the periodic inspections and get a Certificate of Occupancy.
All of this is possible only IF you are the owner of the property.
The powers that be have decided, many years ago, that the process of designing and building one's primary shelter, a house, should not legally involve professional services of any kind.
An exception to this is typically found only in gated communities for some reason, although I think it could be challenged in court. My guess is that when you write a purchase contract for a lot in an exclusive enclave you are also agreeing to the design covenants which stipulate an architect must be involved. And a licensed contractor.
Many states do not require licensed architects or builders in order to design and build single-family dwellings.
Architects MUST be engaged for any other public structure such as a restaurant, office building, commercial retail center, fire station, etc. And typically only a licensed contractor is allowed to build a public structure.
(I am not an attorney, and there may be exceptions to the statements above.)
Let's say creating a set of construction drawings is not your forte but you'd like to save as much money as possible on the construction. From sufficient basic drawings, you can pull your own permits, find subs to bid the various parts, and oversee the construction. You could hire a construction manager to assist for example.
Why engage an architect or other 'designer' to create the drawings?
A draftsperson working for an architect or builder will know exactly what to include on the Permit Set and can take a rough sketch to finished drawings. You could develop a schematic plan with a consumer CAD program like the one described in this recent blog. The draftsperson can point out any discrepancies and rework it for you.
What does an architect have to offer vs. a person trained by experience but has no formal education?
Architects must have a 5-year minimum degree to qualify for licensing. During those five years they learn how to draw conceptually and to scale, how to analyze space, how to put together complex spaces in a logical pattern for living, how to interpret difficult client programs into a workable and buildable project, how to work with legal contracts, to understand environmental systems, to understand and calculate structural systems, to be aware of current technologies and applications, to have studied the history of building through the ages and identify stylistic and technological periods with their merits.
University students might design ONE house in those 5 years. They are prepared to design commercial structures for public use: large multifamily complexes, entire mixed-use areas that combine single and multifamily housing, restaurants, and office buildings into one complex.
Upon receiving their Bachelor Degree (5 years) they are qualified to take a licensing exam, after spending one to two years as an 'apprentice' documenting their work.
If they pass, they can call themselves an 'architect'. No other person may use the word architect unless they have qualified for a license by passing this two-day exam and having graduated from an accredited university.
During these approximately 7 years of preparation, others who have wanted to design houses but sought no formal education have spent the same amount of time working for builders, other non-licensed firms, or architects learning the basic drawings for construction. These people have had no other formal background. They may interact with clients and under the tutelage of a knowledgeable professional may develop the skills to create very good houses.
So an architect just out of school and recently licensed probably cannot design houses as easily and correctly as his or her alternate who has spent the same amount of time working directly and solely in the home design business.
In the last 10 years or so, the unlicensed group has formed one or two associations that administer a test of their own to qualify as a 'professional' with proper knowledge in the design of houses. One such organization is the A.I.B.D. This is the American Institute of Building Designers. Their professional counterpart is the A.I.A. or American Institute of Architects. The A.I.A. in association with other academic organizations and manufacturers mandate Continuing Education courses, represent all members nationally and engage political groups and other organizations in order to improve their membership and qualifications.
If you engage the services of others to design your home, credentials are important. But their qualifications and success are based on years of experience in this house specialty. Look for the bright stars if you undergo this route.
NOTE: I am not an attorney! Please check your local codes and governing bodies to check the information presented above.