Trade mark protection is territorial in nature. This means that, provided all conditions are met, such protection will be granted to your name only in the territories in which you have registered it as a trade mark. In other words, registering your name as a European Union Trade Mark only gives you exclusive rights to use that name with regard to specific goods and services (the ones identified in your application) across the European Union only.
For this reason, it is always advisable to develop your trade mark protection strategy in line with your business strategy – that is, to seek trade mark protection in all the countries where you intend to commercialise your products.
If you do not do so, your name or a similar one may be registered by third parties in those countries, possibly for similar or identical goods. Since you will not have exclusive trade mark rights over this name in these jurisdictions, others will indeed be allowed to protect it via trade mark. This might lead consumers in this area to mistake someone else’s products for yours. Commercialising your own product in those jurisdictions under “your” name, but without securing exclusive rights to it, might even in this context constitute an infringement of those third parties’ rights.
Furthermore, even if no-one else registers a name similar or identical to your trade mark in those countries, not having a registered trade mark there means that you will not have any intellectual property rights to that name in these territories. If other entities (such as competitors) copy your products and sell them on these markets under “your” name, you will not be able to enforce any rights (since you will not have any such rights in those territories) and in particular will not be able to exercise anti-counterfeiting actions.