As others have pointed out, it's not stable to have such an arrangement. However, if your lake or sea is moderately large and positioned correctly then there is the option of having intermittent rivers on each side driven by tides. That particular configuration requires a lake/sea spanning a significant fraction of the globe, a huge influx of water, and isn't geologically stable.

Very narrow straits can also have major tidal currents and if you get two opposite openings into a smallish sea then there will be a rushing river effect twice a day at each end. The straits will open over time, though, due to the erosive power of the moving water. The Mediterranean has such an effect but the channel there is huge.

Any lake or sea is in equilibium between inflow, outflow, and evaporation. Consider what happens if you have a small opening between the ocean and an inland sea. Evaporation in the basin will cause local sea level to drop, making the sea smaller. If the basin is large enough and the openings small enough, then the sea inflow will form "rivers" that go in channels to a hypersaline lake/sea at the bottom of the basin.
This sort of thing has happened to the Mediterranean and Black seas during geologic history. The bottoms of the basins are well below sea level and incredibly hot (dener air at the bottom helps to retain more heat). Freshwater rivers flow from the mountains into the basins to the small sea at the bottom, giving nice oases in the potentially nasty desert. Saltwater rivers flow from the ocean channels to the bottom of the basin, providing no drinkable water. It's certainly not like any conditions we have today (yes, the Afar region of Africa has a basin with saltwater streams trickling from the ocean, but it's not big enough to get the full effect).
This sort of arrangement is not geologically stable, of course, because once the sea breaches the basin walls then the erosive power of a water flow with a few thousand feet of head takes over and your basins tend to fill within a few thousand years (or much quicker if there's a catastrophic breach as happened in the Black sea). It's a great lost civilization story, of course (think Noah).