No it won't.
Ultimately, your whole house will smell bad.
Nope.
Don't believe it? Have your plumber haul off your toilets and just do your biz in those open pipes. Whew!
That's a false comparison, since by removing the toilet you'd be removing the built-in "trap" as well. That's an invalid comparison to the shower, since shower drains *do* have traps, they aren't just open pipes, as a toiletless toilet drain would be.
Modern toilets are designed to keep the smell from escaping into the house.
And modern shower drains are also designed to keep smell from escaping into the house -- if they weren't, gasses from the community sewer line could enter the house through the shower drain:
Ironically for this thread, the plumbing term for that device is a "P-Trap"...
Pee in your shower and you might as well throw your toilets out!
Your conclusion does not follow.