It doesn't make grammatical sense that "אני שוחה וכואב לי" would mean "I swim and it hurts me" or "I swim and it hurts" because you have "I" as the only subject stated and then you have two verbs connected by the conjunction "and" which are "swim" and "hurt". That means that if you separate the conjunction, it's "I swim" and "I hurt". And then you have the direct object "to me" at the end which could either be connected to just the last verb, or to both verbs. So the most literal translation is "I swim and hurt me" Meaning either, "I swim me and I hurt me" or "I swim and I hurt me." Swimming is grammatically unable to do the hurting without the use of a pronoun. I really do not like this sentence if it's really the way they say it in Hebrew.
Actually the Hebrew sentence subtly suggests that something else causes the hurting, because if it were the swimming it would be a bit more natural to say אני שוחה וזה כואב לי. Not using זה, and still using it in conjunction with "I swim" makes me expect this sentence in a context where it would mean "It hurts despite me swimming".
I'm still not sure what happened to the "לי" here, but it is perfectly reasonable to translate the rest as "I am swimming and it hurts me" because in english (and, I'm sure, plenty of other languages) verbs can serve more functions than just as pure verbs. A verb can stand in as a noun as well as a verb (it is called a gerund in that instance), so "it" is the swimming. "I am swimming (verb) and the swimming ("it"; gerund) hurts." Because of the nature of the Hebrew verbal case system, a pronoun of any sort can be superfluous unless context requires you to specify what the "he/she/it/they" is for clarity's sake. In order to say this with greater precision, it would need to say "I am swimming and the swimming hurts," which to those "in the know" ( which isn't me; I know general grammatical structures, but not how Hebrew itself applies them), may very well seem ridiculous.
(It's not possessive, it's pronominal indirect object or something...)
In English you can say "it hurts" and leave it to understand from the context who feels the pain. In Hebrew it's strange: you could say זה כואב and not specify the object, like in English, and it would sound fine. Or you can omit the זה, but then it doesn't sound normal to say just כואב. If you add the object it suddenly sounds fine again: כואב לי.
I wrote the same as Patar595: I swim and I hurt, although my firs though was:I swim and it hurts me. But I still wonder: Is it obvious that the pain is caused by the swimming or could the swimming be one and the hurting just stating another fact, example: My hart was broken because of all the bad news, but I still went to the beach, and now I am svimming and hurting......Or my foot was injured cause I stepped on a piece of broken glas as I entered the water...swimming and hurting...