If it ain't raining our policy is just strap and roll. I've got one OSB mill that I go to that won't let you leave without tarping if there's even the slightest chance of rain. Other than that, everyone else has the "once it's on your truck, it's not my problem" mind set.
I'm in a day cab and on a dedicated account so I deal with the same people all the time. I go all over NC, SC, VA, and parts of TN but I see the same faces a few times a month and have built a decent relationship with all of them. That helps a lot when it comes to handling issues that may arise.