2 people can not open the same document.
Because, I am assuming by your use of "word" that you are not talking about the word application, but a word file instead.
Opening the word application does not open a word file, it just brings you to ablank word doc. And each person should have word installed locally on their machine.
But If you are talking about having a 3rd party appliaction opening the word appliaction that is installed on a central server. And that third party app allowsyou to utilize the word app installed on the server. Then it is the 3rd party app that is messing up. Not word.
Because if you are using a thin client tool such as Citrix/MS Terminal Services, which allows mulitple people to use the same program that is installed on the server. Then this should not be an issue.
But if your clients machines word exe points to the word exe that is installed on the server instead of the one that is installed locally, then there could be a problem.
Justto be clear though, 2 people can not open the same word document at the same time. It will throw a lock on that file.