If BW Widgets registers the mark "BestWidgets" in country A, no one ease may register the mark (dore widgets) in country A, and anyone who uses it (to sell widgets) in country A is subject to an infringement lawsuit. However, GR widgets can register and use the mark "BestWidgets" in country B, and thew registration in Country A will have no effect on this at all. If BW Widgets wants the mark protected world wide, it would need to register in every country. That would be costly.
Note that in many countries, a trademark gets no protection at all unless it is properly registered in that country. In the US, more limited protection is given to marks actually used in trade but not (or not yet) registered.
Note also that marks are specific to an industry or type of product. If "BestWidgets" is registered in country A as a mark for a kind of computer product, no one else can safely use that to sell computer products without permission. But if someone wants to use that mas the name of a brand of fruit salad, the registration would not stop them.
Use of the "tm" symbol has no legal significance, and anyone may chose to use it at any time. By custom it indicates a trademark claim that is not currently registered, and provides some notice of intent to register or claim the mark.
Use of the ® (r in a circle) symbol is subject to regulation. In the united-states only marks currently in the Federal register may be labeled with this symbol. It is not required that it be used, but using it makes establishing some claims easier. See "When and How Do I Have to Use Trademark Symbols?" from Forbes for more on this. Other countries have similar laws.
If "BestWidgets" is registered by BW Widgits in in country A, and BW Widgits has a web site hosted in and targeted at A, it could use an ® symbol with that mark on that site. If GR Widgets has registered the same mark in country B, it could use an ® symbol with that mark on a site hosted in and targeted at country B. In each case a note somewhere on the site indicting in what country (or countries) the company claims the mark would be a good idea, and would make it harder for the other company to sue claiming infringement in its home country. Such a suit might be brought, and how it would end up would depend on the details of the facts, including the market for widgets.
A domain name is not automatically tied to a trademark. The registration process is quite separate. There are cases of so-called "domain squatters" -- people who register as a domain name a more or less well-known trademark, in hopes of selling the domain to the trademark owner. Domain registrars have procedures by which a trademark owner can assert that a domain is registered to a squatter, and seek to claim it. Evidence that the person who has registered the domain holds the trademark in some country and does actual business under that mark in that country would tend to defeat such a claim, but each registrar has its own procedures.