How are import statements in plpython handled?

The declaration in the body of a PL/Python function will eventually become an ordinary Python function and will thus behave as such. When a Python function imports a module for the first time the module is cached in the sys.modules dictionary (https://docs.python.org/3/reference/import.html#the-module-cache). Subsequent imports of the same module will simply bind the import name to the module object found in the dictionary. In a sense, what I'm saying may cast some doubt on the usefulness of the tip given in the accepted answer, since it makes it somewhat redundant, as Python already does a similar caching for you.

To sum things up, I'd say that if you import in the standard way of simply using the import or from [...] import constructs, then you need not worry about repeated imports, in functions or otherwise, Python has got you covered.

On the other hand, Python allows you to bypass its native import semantics and to implement your own (with the __import__() function and importlib module). If this is what you're doing, maybe you should review what's available in the toolbox (https://docs.python.org/3/reference/import.html).

The import is executed on every function call. This is the same behavior you would get if you wrote a normal Python module with the import statement inside a function body as oppposed to at the module level.

Yes, this will affect performance.

You can work around this by caching your imports like this:

CREATE FUNCTION test() RETURNS text
LANGUAGE plpythonu
AS $$
if 'json' in SD:
    json = SD['json']
else:
    import json
    SD['json'] = json

 return json.dumps(...)
$$;

This is admittedly not very pretty, and better ways to do this are being discussed, but they won't happen before PostgreSQL 9.4.