The Cambridge Dictionary offers this explanation of obsess:
If something or someone obsesses you, or if you obsess about something or someone, you think about it, him, or her all the time
What's unusual about this word is that, for the intransitive form, the subject is the person, and for the transitive form it's the thing that they think about all the time.
A lot of young girls are obsessed by their weight.
Looking at the first of the sentences that you quoted, the agent (doer) is clearly the weight, and this is perfectly OK according to the definition above.
Their weight obsesses a lot of young girls.
It is possible to convert this sentence to active voice, but this is an example where passive voice works better: first, because the most important thing in the sentence is the girls, and second, because it's strange to start a sentence with a possessive "their", before the girls themselves have been mentioned.
Jody’s been obsessed with some lifeguard for months.
Looking at the second sentence, with seems to be an unusual way to attach the agent, but in the past this was more common. Here is an example:
Mankind are still possessed with all the foul passions in the catalogue of sin. " - Letter of a village governess, Elizabeth Bond, 1814
As this NGram graph shows, the with version is used a lot more frequently than the by version. This is probably understandable because, although the lifeguard is grammatically the agent in this sentence, the person who is doing the thinking is Jody.