Seattle is swimmable. Lake Washington, Green Lake (both fresh water), and the Puget Sound (salt water) are swimmable in city limits. I swim in lake Washington 3 times a week May - October.
As in another comment here about Copenhagen, you cannot swim after hard rain because of how the sewer systems were designed. Seattle is famous for rain, but not the type of rain that will cause a spill over event often.
The county government also tests the water for bacteria and algae and reports the findings on their website and closures are signed at access points.
https://kingcounty.gov/en/dept/dnrp/nature-recreation/parks-...
Also while there is a lot of private ownership of water access, there are 9 city beaches and something called street ends, which are public right of ways if a street ends at water, I swim from these all the time.