I think the connotation you are looking for really depends on the context of these sentences. Neither one has a strong enough connotation on its own to make the assumption you are about what it implies.
For example, compare:
He lives by himself now. He can do what he wants when he pleases. There is no one to tell him when to eat, when to sleep. For the first time in his life, he has privacy and he loves it.
His wife left in January. He lives by himself now. For the first few months, he secretly cried every night. Even now, sometimes he comes home to the cold, dark house and wonders where it all went wrong.
The sentence works in either context. Likewise, living on his own could very well be positive:
Shortly after he started walking as a toddler, he was diagnosed with autism. His parents were sure he'd never be able to live independently, but after years of successful therapy, he lives on his own. They are so proud of him.
The context is really going to affect how you interpret either of these sentences.