I'm very uncomfortable with the idea of trying to negotiate a construction salary, but everyone talks about it like it is normal. How often do construction employers really allow negotiations?

It really isn't a matter of allowing. It's a matter of whether you know what you're worth and whether you believe you are worth it. Some surveys say that as many as 65% of the hiring construction managers are willing to negotiate and have planned their initial offer to allow room to move upward. Some make take-it-or-leave-it offers. So, the bottom line is what are you willing to take and what will you walk away from. Know that, and you are in a position to negotiate.