Every EU competition that EPSO publishes on its external website is by default open to anyone who meets the formal criteria, including EU officials who may wish to apply to a higher grade, or temporary agents who wish to "finalise" their contract.
On the other hand, EPSO occasionally (usually every 5 years) organises internal competitions which are only open to those already working as permanent or temporary staff at the Commission or other institutions, depending on the exam's specific requirements.
See also:
EPSO | EU Commission | Contract Agent | Temporary Staff | Notice of Competition | Internal Competition | Staff Regulations
Related link:
Selection and Recruitment | EU Recruitment System A to Z | EU Jobs