If you can afford it, I'd do both - play for the team (or another team) and train on your own. I'm sure it's possible, but I'm not sure I've heard a success story where a kid elected to not play and just trained. Remember baseball's a game. Meant to be played and enjoyed. Remove the fun and joy elements and it'll take a VERY special player to stay motivated all summer enough to actually improve and grow.
Way off topic, but I think it's a decent metaphor. I once saw an interview with 80s pop star Rick Springfield where he was asked what he does when he isn't inspired and can't write songs he likes. He said "I give myself permission to write sh$%y songs." He keeps working. Grinds through it. Suffers some, but he never stops working. I liked that answer.