This is determined by a country by country basis and different countries have different laws.
In Australia we us ASIC and IPA Australia to look up specific trademarks. All CC TLDs (Country Code Top Level Domains) are governed by auDa - the .au Domain Administration.
Australia has very specific compliance for our CC TLDs, including trademark found in
Domain Name Eligibility and Allocation Policy Rules for the Open 2LDs. This covers common misspellings of a trademark termed and you may submit a complaint via the appropriate channels via auDa.
This is a common approach with most ccTLDs having some type of Governing body to self-regulate domain names and intellectual property.
Where it gets complicated is with gTLDs (Generic Top Level Domains) such as a .com, .net, .org or countless others.
This is on a country by country basis and in many instances if a business is trading in the same country as you, does not have the trademark and you do then you would probably have to write a letter exclaiming infringement.
Generally these matter are settled with some kind of arrangement or a negotiated settlement. But it can be complicated - if they own the trademark in another nation, they do not trade in your country, your protection laws in your country do not allow protection on anything beyond ccTLDs etc.
Situations like that are what lawyers are for.