Effect of finite speed of light on timings of events In relativity when an observer records an event some distance away from him, the timing of the event he measures would be different from what he would measure if he was close to the event as light would take some time to reach him. So when we say that he observes the event at time t(say) then is it according to what a different observer would measure who is at the same location as the event and whose clock runs like that of the first observer?
 A: 
In relativity when an observer records an event some distance away
  from him, the timing of the event he measures would be different from
  what he would measure if he was close to the event as light would take
  some time to reach him.

This isn't true in Special Relativity (SR) - the observed time of an event is the time recorded by a (synchronized) clock at the spatial location of the event, and it is not the time that light from the event reaches the origin.
That is, observing is the not the same as seeing (or photographing) in SR. From the Wikipedia article Observer (special relativity):

The special relativity usage differs significantly from the ordinary
  English meaning of "observer". Reference frames are inherently
  nonlocal constructs, covering all of space and time or a nontrivial
  part of it; thus it does not make sense to speak of an observer (in
  the special relativistic sense) having a location. Also, an inertial
  observer cannot accelerate at a later time, nor can an accelerating
  observer stop accelerating.
Physicists use the term "observer" as shorthand for a specific
  reference frame from which a set of objects or events is being
  measured. Speaking of an observer in special relativity is not
  specifically hypothesizing an individual person who is experiencing
  events, but rather it is a particular mathematical context which
  objects and events are to be evaluated from. The effects of special
  relativity occur whether or not there is a sentient being within the
  inertial reference frame to witness them.

In summary, an (inertial) observer in SR is not a being with a definite location but is, conceptually, a system of unaccelerated rods and (synchronized) clocks at relative rest that assign space and time coordinates to each event. 
A: I would expect that most observers would take into account the speed it takes the light to reach them, when they measure the time of an event. So, two observers with zero relative velocity to each other, but at different distances from an event, would measure the same occurrence time for the event (assuming they have synchronized clocks).
