Each class gets its own set of attribute points. You can go ahead and put all your Pugilist points into strength and still be able to put all your Arcanist points into intelligence for your Summoner. Arcanist is actually the only somewhat tricky one in that regard. Because jobs are specialties of their base class, they use the base class's attribute points. Since ACN can become both Summoner (who wants Intelligence) and Scholar (who wants Mind), you'll have a decision to make there about which to favor if you want to follow both of those jobs, or just switch back and forth.

Lots of people do many or all classes and jobs on the same character. It's perfectly viable and in fact has some advantages in getting cross-class skills and in collecting gear. When people (like me) do multiple characters, it's usually for more casual (fluff?) reasons like wanting different looking characters of different races, or in order to do the story repeatedly.