If your document is returned from the designer looking exactly like your Word document then you're using the wrong designer. If they can't produce anything better than Word and they have InDesign, well...
Truth be told, and never mind the poorly formatted sentence that follows, this answer depends on what you're using it for, how much your return is from these documents, and how big your audience is. For example, are you spending $10,000 a year on your designer to copy your Word document and paste it into InDesign and your return is only $5,000? If so, you need to rethink your strategy. Are ten people seeing this? Is it for in-office documents?
If any of these answers are yes then maybe sticking with Word is your best strategy. InDesign is for very professional layout design that gives the user full control over every aspect of the document. Word is meant for the lowest common denominator and is meant for the average user to do what Word has decided they should be doing. There are a lot of features in InDesign that are just controlled by Word that not only make for a good-looking document, but also a proper print job. Really, you shouldn't even worry about printers unless your market is big enough plus your return. When you ask about what "most companies" are doing, that depends on those factors. Many companies are using fly-by-night designers and bargain basement printers for all their stuff. Many other companies hire professional designers and consider quality printers over discount printers. Still many other companies have a designer on staff who does all of their work in-house. These are most often, in order, small, medium, and large companies. Find your tier and you will find your answer.