My online games are typically only an adventure or a few strung together. Typicaly not enough time passes that skills would be gained.
ShawnDriscoll said:
role-play their training.
For our in person campaign, skills gained are based on role play and not just the characters saying "in my spare time I'm...".
For us,there is actual schooling since we have played military characters or characters that work for the government and sometimes they get a training assignment. This is often several months of doing nothing but learn from professional instructors.
It's typical for crew on our ships to cross train. The doctor was so proud of what she had learned about engineering and would tell everyone who came aboard how she helped install the new plumbing and wiring when our common area was remodeled.
At some point the GM might decide players have done enough to warrant an increase in skill or a new level 0 skill.