Do I have to have a "localised" domain name?

No. In Australia for instance, it is unfortunately common belief that Companies and Businesses have to register a name which is derived from the company or business name in the same order of syntax as their trading or registered business name. And that it must have the country extension following the "dot com , that is .com.au. It is because of a lack of understanding that Businesses don't realise that they can break the shackles of being stuck with a poorly worded local domain name and just get themselves a better name without the restrictions of a localised domain name extension.

A hypothetical example would be "Joseph Mandianna & Sons Cooking Spices Pty. Limited". If Joe was to register a domain name in the localised form it would have to be something like "www.jmandiannacookingspices.com.au" or "www.jmspices.com.au" or "www.jmandsons.com.au" etc. etc. Very restrictive and it also means Joe is likely to only get to sell his spices on the local market. That may be fine for Joe if he is happy with just selling locally because no-one from outside of Australia is going to find him. The "dot au" extension may well be completely overlooked on the major global search engines. If Joe decided however that he really would like to trade internationally then he would be well advised to get a global name as well. He could keep his "local" domain name as the name of his website but then he could also register or buy a more succinct "Global" domain name or names that also point to his site. "www.cookingspices.com" would certainly get Joe a lot more traffic and increase his bottom line accordingly. So Joe has his local domain name submitted to the localised search engines driving the local traffic, and he also has his "dot com" submitted on the "global" search engines bringing in overseas traffic.

Makes for good business.