Candidates who hold the nationality of any European Economic Area (EEA that includes EU, Iceland, Norway, and Liechtenstein) member state or Switzerland can easily be employed by an Estonian company. No residence permit or work permit is required. A thing you have to do before the employment contract enters into force is to register him/her in the Employment register.
This employee working remotely does not become an Estonian tax resident, therefore your company is free from tax reporting and payments in Estonia. However, your company may be obliged to declare taxes in the country of residence of the employee and be registered as a social taxpayer
. That’s how it works in France, Italy, Spain, Germany, for instance. In several countries, you may need to open a local branch of your company. Anyway, you will pay payroll taxes in accordance with local tax rates.