The following versions are syntactically unambiguous and grammatically correct. In different contexts each can be semantically correct (factual): the first two are factual in a context where President and host are one person; else the latter two are factual.
The President and host greets the guests.
The President and the host greets the guests.
The President and host greet the guests.
The President and the host greet the guests.
In spite of the first two versions being "correct", they are quite likely to be misunderstood. After being led up the garden path by an apparently-plural subject, one hits the rock of a singular verb. Is it one's first response to say, "Oh, I misunderstood, they meant the President is the host"? Of course not; the first thought to mind is "Oh, look at this subject-verb disagreement; this writer's illiterate." The following avoid that problem. Of course many other rewordings exist that may better suit a given context.
As host, the President greets the guests.
The host (the President) greets the guests.