For Example : Can the owner of Twitter.com and Twitter.me accuse each others for the same company name?

so twitter.com has their own company , logo idea of their products ( Web apps , Mobile apps ) and twitter.me has their own company , logo idea of their products ( Web apps , Mobile apps ) in the same country so is there any kid of problems ? we can say they are all have trademarks or do not .. so how we can settle this up ? cause i have seen many other examples of same name of apps same company name and maybe same country so is that fine ! and the only matter is diffrent domain suffix so can i have twitter.info and have kinda same company name and same name for web and apps ? it will be different idea and different content different logo but what i share with the other companies is only the name of " Twitter " or similar domain name ...all of us not do any sells or goods only websites