Typical deterministic game engines will do this, send it to every machine as part of the initial game state, and also check the seed across machines on every simulation frame (or periodically) to detect desyncs.

That could lead to other subtle problems elsewhere though, because it requires synchronizing the seed. If you can't do that, it could lead to problems. E.g. when comparing offline speedruns where everyone would have a different seed. Then some players could have more luck than others even with the same inputs, which would be unfair. (Though I can't think of anything else at the moment.)