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Not hard to catch me out on Saturday

The setup on the experiment isn't very good because technically the particle is going to be observed flying at observer 2 at one speed and then moving away at another (Plays stereotypical racing car noise of approach an recede).

However ignoring you bad setup the formula for time dilation for observer 2 is

dilation time = time / sqrt(1-(v2/c2))

plugging in

dilation time = 1s / sqrt(1-((0.8*0.8)/(1*1)))

dilation time = 1s / 0.36 = 1.6667 sec

So observer 2 sees the particle take 1.6667sec emitter to detector (ignoring bad setup)

The other way of saying this is observer 2 would say the particle was doing 0.54c not 0.9c that observer 1 is seeing, so he is seeing it redshifted by quite a way.