A PEO shares the employer relationship with you. Co-employment means the PEO is the legal employer for tax, benefits, and workers' comp purposes; you stay the worksite employer who hires, fires, and manages. The structure works within the US — every US state has its own labor laws, but PEOs operate across all 50.
An EOR (Employer of Record) becomes the sole legal employer in the country or state where it operates. There's no co-employment — the EOR's legal entity in France employs your French engineer; you don't have a French entity at all. The EOR handles French payroll, taxes, statutory benefits, and labor law compliance under its name; you direct the engineer's work but have no direct employer relationship.
The distinction matters operationally. Under co-employment, your company's name still shows up on most employment paperwork (W-2s show "PEO + your company"). Under sole employment via EOR, the EOR's name is on everything — your employee is technically employed by Deel or Remote or Velocity Global, not by you.