Swimming the Aare?

I don't know how it is done in Bern but in Basel swimmers have a floaty waterproof bag that is attached to their arm. The bag has everything the swimmer needs, nice and dry.

most people shown would be locals, their "life essentials" are left at home. Bern is a small place only 5 miles across so home would be within walking distance.

There are tour companies that will provide you with swimming gear and transportation.

I will look into tour companies, thank you. Have you hired one?

At least in Basel, waterproof bags shaped like fishes are sold at the Coop markets. They keep your stuff dry and double as a bit of a floatie and a way for ships to see you if you're in the heavily trafficked Rhine. I don't know if they're sold elsewhere but you might be able to find one - or another waterproof bag- online.

Swimming the Rhine was the most memorable experience I had in Basel and wouldn't hesitate to jump in any Swiss River that's safe.

Not just Coop sells them lots of shops do. The are called "Wickelfisch" (=changing fish, as in changing into your swimming gear).
info here: https://www.basel.com/en/Basel-is-an-inspiration/Summer-in-Basel/The-Rhine-is-Life
Buy them here (Migros): http://www.sportxx.ch/de/ausruestung/trekking-zubehoer/reisen/wickelfisch-wasserdichter-badesack-gross-wickelfisch/pp.491075900000

The swimming season is when the temperature gets above about 25°C, June to early September. Last weekend it was ~31°C and lots of swimmers, today it was 23°C and I saw nobody.

"Life's essentials" would be clothes, ID, a bit of cash and public transport ticket to get you home, all of which are safe in the Wickelfisch. I don't see why you need your suitcase with you to go swimming!

In Basel swimming in the Rhine involves as much floating as swimming. Due to the current you always come out downstream of where you went in, and walk back along the riverside path. There are showers provided on the path to allow you to clean off.
There is a poster showing the recommended places to swim: http://www.polizei.bs.ch/dam/jcr:8057f130-eb5e-48f5-92e4-35f4931ec998/Schwimmplakat%20Wegweiser%202016.pdf

Try googling "Wickelfisch", I even found a site importing them into the USA.

Good info Chris! I treasure my Wickelfisch as a practical souvenir and was kicking myself for not bringing it on a trip to the Altmuehl river in Fraconia last weekend! I could have used it!

There is a free public outdoor swimming pool (Freibad Marzili) adjacent the Aare from which you can access the river. There are lockers you can rent to keep a change of clothes, valuables, etc. You can follow a path up river to choose where you want to enter the river. About a kilometer or two up the path is a footbridge a lot of people jump off into the Aare and float back down to exit at Freibad Marzili.