Use "in" instead of "on". Or write "There are 2 schools on the southeast side of the area." I'd do the first with "in". Southwest would be used over south-east in must situations.

Here is my advice on writing style. 1. Use "two". 2. Use "in the south-east". (see Oxford Dictionary) 3. Use the proper specific description of the place with a clearer sense of boundary such as "district", "county", "province", "estate", "economic zone", "town", or "city", rather than a loose noun such as "area" if you are invoking something geographically specific like "the south-east".

Neither is correct, actually. The preposition is wrong. Between southeast and south-east, you need to check a style guide. Moving from nouns to a hyphen to a single word really depends on the evolution of the expression. Sorry, there's no simple answer here. This is where English changes quite quickly.