For these examples, several correct punctuations are possible besides the correct one shown in your first example. In the second one, the comma and semicolon combination is wrong. It would still be wrong if one to the east were replaced by an independent clause, as then you'd have a runon sentence. It's less clear-cut when one to the west is replaced; for example, It had two entrances, one of them to the east; and that's where I went in is sound.
As noted, the first example is correct, but it's portentous (ie puffed up). Instead of The construction site had two entrances: one to the east, and one to the west I'd write The construction site had entrances east and west or perhaps The construction site had east and west entrances, and let people assume no other entrances exist. But if it's important to emphasize the existence of no other entrances, there might be nothing much shorter than what you have. For example, I think neither of The construction site had two entrances, east and west and The construction site had two entrances, both east and west perfectly distinguishes between there being one east entrance and one west entrance vs two east entrances and two west entrances. The construction site had just an east and a west entrance might work, or more explicitly, The construction site had just one east and one west entrance.