2-Layer PCB Through-Hole Layout/Soldering
Plated through-holes have metal not only on the far side and near side, but also on the inner edge of the hole all the way through the board. Only on single-sided boards or boards made by unconventional methods (such as milling) will you see pads that aren't plated on the inside of the hole.
Of course, this is all assuming your drill files specify the holes as plated. If you specify a hole as non-plated, then they won't plate it. This is down to how you use your CAD software, but unless you're doing things very wrong with making footprints, all holes where electrical connections are made should be plated through.

From the image you gave, it looks like you're using KiCAD; in KiCAD every through-hole pad is either a PTH pad (standard through-hole with plating on it) or an NPTH pad (which is referred to as &quot;NPTH (mechanical)&quot; in the pad editor, I believe), and this is a property of the pad that you select when making the footprint. All the footprints included in a standard KiCAD installation have the pads appropriately selected; PTH for all the ones that need an electrical connection or where you solder in a mounting pin, and NPTH for all the ones used solely for mechanical mounting with rivets, screws, plastic bumps, or whatever else you may need for a given package.