The rule for businesses

The European Commission says economic operators established in the EU customs territory need an EORI number. Operators not established in the EU need one when they lodge customs declarations, entry or exit summary declarations or temporary storage declarations, or when they act as carriers (European Commission).

The rule for private persons

There is no single EU answer; customs authorities apply it nationally. German customs say private individuals do not need an EORI number if they lodge declarations only occasionally, fewer than ten times a year, and that courier services acting for private customers have no EORI obligation either. For exports that need a licence, a private person may still have to give one (Zoll). Finnish customs use the personal ID for private import declarations instead (Tulli).

How the check decides

Business established in the EU: yes.
Business outside the EU: yes once you lodge declarations or act as a carrier; otherwise not yet.
Private person with a licensed export: possibly.
Private person with ten or more declarations a year: check with customs, based on the German threshold.
Otherwise, for private persons: normally no, though a seller or carrier may still ask for a number.

The slider shows your yearly declarations against the German threshold of ten. Other countries may draw the line differently.

Next step

If the answer is yes, see the application country check, then who needs an EORI for the full rules.