You seem to be thinking about the "pair production picture" of Hawking radiation, which states that the Hawking effect is due to pairs of virtual particles being created near the event horizon, one of them falls in the black hole, the other one escapes, and we see the escaping particle as radiation. That is a pictorial description, but it is intrinsically wrong and should not be taken too seriously.
What happens in both the Unruh effect and the Hawking effect is that the notion of particle is observer dependent. What a particle is does depend on who is measuring it. The particles seem by accelerated observers in Minkowski spacetime do not come from pair production or anything of the sort. The state simply has particles when seen by an accelerated observer.
Hawking radiation is a little bit different. In the Hawking effect, one has a spacetime that starts with a star, the star collapses to form a black hole, and the black hole eventually settles down. What happens in here is that the notion of particle in the "star era" of the spacetime (AKA the asymptotic past) does not correspond to the notion of particle in the "black hole era" (by which I mean the asymptotic future). Hence, a stationary observer in the spacetime sees particle creation due to the fact that the very notion of what is a particle has changed.
The key point is understanding that these effects concern the quantum theory of fields, not of particles. Particles are not a fundamental notion in Physics, fields are. Particles simply arise as convenient interpretations of certain quantum field states, but they do not play a fundamental role in the theory. The particles in the Unruh effect do not arise from pairs or anything of the sort, it just happens that different observers call different things by the word "particle". This is similar to how different observers have different notions of time.
If it interests you, I've given a more detailed description of these processes in this answer, which still attempts to not dive too deep into the math of QFTCS.