Can a PCB circuit have through hole components on both sides of the board
There is no PCB design or electrical reason this is not feasible and in most cases you would need to define a custom DRC for this to be an error. Note that unless you have appropriate 3d models interference between TH parts won't be checked either.

With two sided TH components, generally speaking at least  one side will have to be soldered by hand for commercial production.  So if you persue commercial production with a contract PCBA assembler or with your own production line you will likely pay much more per board to manufacture . With one sided TH it can be wave soldered , greatly reducing cost. 

If you are assembling small runs by hand anyeay, or designing as a kit, this may make things more difficult, but generally not as big a problem.