EMIR doesn’t have many extraterritorial demands when it comes to reporting aspects (and ESMA is just the regulator policing EMIR and other EU financial regulation). If you are a non-EEA entity you have no reporting responsibilities under EMIR and if an EU entity is trading with you they are able to report a one-sided trade from their perspective.
ESMA has expressed the preference that non-EEA counter parties obtain Legal Entity Identifiers but have no way of enforcing this. However, when it comes to EMIR clearing obligations there is a little wrinkle which could affect non-EEA firms.
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If you are a category of firm which would be subject to a mandatory clearing obligation if you were in the EU, then that trade will be considered subject to mandatory clearing anyway, and either both parties will need to submit their side of the transaction to an authorised CCP, or the EU party will need to submit both sides. This is likely to cause some falling out among international trading partners (but in the short term will be confined to large Interest Rate and Credit Default Swaps contracts).