Dedicated or server with just one of the PC's playing the game, you can usually specify if you want it LAN only, or if you want it open to the Internet. Most servers if you don't open a port on your firewall and forward it to the right computer it wouldn't work online anyway.
Broadband speed does become an issue if you are hosting a server for more than a few people coming in over the Internet, each game is different as to how many will work for your connection. So yeah it can become laggy if you have too many people on, usually for everyone not just the external people (but again every game is different).
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Uncertainty is an uncomfortable position.
But certainty is an absurd one.
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