Omega serial numbers are engraved on the movement on vintage production and on the case or caseback on modern watches. The movement also carries a calibre number, and calibre and reference must agree — a Speedmaster Professional should contain the calibre its reference specifies, and nothing else.
For vintage watches, Omega's archive service will supply an Extract from the Archives confirming the production date, the original calibre and the market the watch was delivered to. It is ordered from the maison for a fee. It is not an authenticity certificate, but a watch whose extract does not match what is in front of you has a serious problem.
Modern watches carry a warranty card and, since the introduction of the maison's chronometer certification programme, a certificate covering the METAS testing. Cards travel independently of watches and prove nothing on their own.