Some companies want you to always drop it at a nearby yard. But I've always taken my truck and trailer home. I park it at a Walmart that has limited truck parking about 2 miles from my place. You'll want to try to find a truck stop to park it in, but if there's none nearby then ask a few businesses around your area (Walmart, K-Mart, Target,etc). You'll usually have the option of taking the trailer home but if there's no place to put it then let your DM know that you cannot take the trailer home and they'll give you the nearest location you can drop it at.
I live about 250 miles from my home terminal so I can take my truck home with me. Generally I am empty and have a truck stop about a mile away that lets me drop my trailer there. I just put my king pin lock on I and bobtail home. It gives me the opportunity to deep clean and re-provision. Plus all the kids in the neighborhood love it and my daughter gets to sleep in the upper bunk which she really digs! (go figure). I wouldn't be able to get a 53' trailer any place near my house. Plus the neighbors would probably have a cow.
"Bobtailing" means you are driving a tractor without a trailer attached.
A facility where trucking companies operate out of, or their "home base" if you will. A lot of major companies have multiple terminals around the country which usually consist of the main office building, a drop lot for trailers, and sometimes a repair shop and wash facilities.
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When you get home time can you take the truck home without the trailer? I would imagine most people just don't have the space for a trailer at the house