When prototyping diy PCBs, we would run a small wire from the underside of the board to the top and solder on both sides, wherever there was connection on both sides of a THT component, so there was good connection on both sides. I am about to order a batch of PCBs from JLC PCB ... do I need to use the same practice or do they coat the layers so that both sides are already making contact on the PCB?
Plated through holes, which is what you seem to be referring to, are normal for both vias and through hole pads in modern double-sided or multilayer PCB fabrication.
Typically this fact would be buried somewhere in the specifications of a board service (there will often also be a limit on the maximum hole size which can be plated, and an indication if slots can be plated or not). Generally a hole surrounded by at least a certain diameter of surface copper will be plated, and one drilled in the absence of a pad or plane will not, but again this falls to the detailed specifications and specific interaction of your software with theirs.
If in doubt, contact the board manufacturer or utilize one with more clearly stated specifications.
I used to prototype PCB's in the same way, with a CNC milling machine and then ordering a regular board.
Do I need to use the same practice or do they coat the layers so that both sides are already making contact on the PCB?
The way that normal PCB's are made is they etch the layers (for a 4 layer board they laminate two more layers on a middle section), then holes are drilled into the board. The holes are then copper plated.
What you need to provide to the board house is the files for the layers and a drill file. The drill file tells them where to drill the holes. The company that manufactures the PCB's will have specifications on how much plating and the minimum size (usually after 10mil (or 15mil for 4 layer) they start charging extra for vias) but everyone is different.