Nationals of EEA member states and Switzerland can take advantage of freedom of movement for workers and do not require a permit to work in the Czech Republic.
To take up gainful employment in the Czech Republic, nationals of countries other than EEA member states or Switzerland (‘third-country nationals’) may need a work permit. Details are provided here: https://www.mzv.cz/jnp/en/information_for_aliens/long_stay_visa/employment_card.html.