The short answer is, “Yes”—it matters very much who you name as your executor of your Will (or Wills) or the personal representative of your estate.
This is one of the factors you should consider in deciding whether to use either (i) two or more single-jurisdictional Wills (one for each country), or (ii) one multi-jurisdictional Will. In particular, one benefit to utilizing separate situs Wills (option (i)) is the use of several executors may ease of access to local counsel familiar with the law within each situs. Each country has its own rules relating to liability of the estate and other taxes. In the United States, an executor or personal representative assumes personal liability by law for the payment of estate taxes. And, where no U.S. executor has been appointed, the Internal Revenue Code imposes that liability directly upon the holder of the decedent’s property.
What this means is that a deemed executor is held to be responsible for the payment of estate tax up to the full extent of the value of the property in his or her possession and will incur personal liability for U.S. estate taxes if he or she distributes assets out to other U.S. beneficiaries, or takes personal possession of an asset before paying U.S. estate taxes. Therefore, it is important to choose an executor who can coordinate with the laws of each jurisdiction or to have multiple executors to do the same. The decision will turn, in part, upon which jurisdiction in which you have your property. Qualified counsel can help you navigate the answer to this question.