Depends how the contracts are written. Private construction firms could be used to do all, part, or none of the work.
Last year, some folks living near the border mentioned that wall construction in some places had already started. There have been stray news stories about the ACoE doing soil sample testing and clearing land.
Trump only needs to build 700 to 900 miles of wall because the border has a lot of dangerous natural barriers. He could probably accomplish this without the public ever noticing. How many news stories do you remember seeing on fence construction that occurred under Clinton through Obama? Probably none, right? They're far too busy with tabloid crap.