He fell out of the rotation towards the end of the season and hardly played at all in the playoffs, which lead to a lot of speculation and rumors about him falling out of favor with the Miami brain-trust. That may or may not be true, although Erik Spoelstra did say he thought Beasley had matured tremendously on and off the court when he was asked about him during the NBA Finals. The reality is that he probably never had much of a chance to get playing time anyway - the Heat were pretty all-in on their veteran leaders in the frontcourt.
Over the course of the playoffs, Miami gave guys like Shane Battier and Udonis Haslem chance after chance, even when it was pretty obvious that they had nothing left in the tank. Maybe Beasley's mental mistakes on the defensive end made it impossible to play him, but Battier and Haslem flat out couldn't move anymore and their physical flaws on the defensive end absolutely killed the Heat throughout the playoffs. It wasn't really exposed until the NBA Finals, though, due to Miami's cakewalk through the Eastern Conference Playoffs.
For all the knocks on him, Beasley proved that he could still play basketball and help an NBA team last season. He's never going to be the elite two-player he was projected to be coming out of college, but he's a 6'8 240 mismatch nightmare coming off the bench and if given the chance to play against second unit forwards on a spread floor, he can get buckets really quickly. If a team is looking for instant offense off the bench from their front-court, Beasley can provide it.
Most of the concern with Beasley stems with his reputation for immaturity, off the court foolishness and run ins with the law. As a result, a lot of fans just don't want him on their team in any circumstance and he has become a punchline throughout most of the NBA. The thing about it is, though, is that he is getting older and as people move deeper into their 20's, they do tend to mature. Maybe Beasley never will, but there's little harm in bringing him in for a workout and talking with him to get a sense of where his head is at.
He's 26, he wants to make a living playing professional basketball and he probably wants to do it in his home country, if at all possible. He knows he's on a really short leash and he's not going to get a long-term guaranteed contract. If you are confident in your locker room culture and you think that he is starting to grow up, there's really no risk in bringing Beasley in on a one-year deal. Once a guy gets a reputation for being a knucklehead, it's very easy to write him off as not worth the trouble, but people can change, even a guy like Super Cool Beas.