Different observers, different fields? Maybe the first part of the question is whether it exists something like "the same point in space" for different observers (not necessarily "at the same moment" if they are moving) although that "common point" has different coordinates for each observer.
The question ends here if the answer to that is negative, as I'm afraid of (since that would imply there is a fixed frame for space).
But, if not: they will be having different values for any field measured at that common point, right?
 A: I think the something you are looking for is an event. That is if one
observer seees something that happens at position $x$, $y$, $z$, and time
$t$, for example a particle is at that position at that time. A primed
observer will observe that event at the transformed position $x'$, $y'$,
$z'$, $t'$.  Assuming special relativity, i.e. not general relativity,
the primed coordinates are given by Lorentz transforming the event from
one coordinate system to the other.
If the observers measure a field at that event, the result depends on
what type of field it is. If it is a Lorentz scalar field, they measure
the same values. If it is a vector field the field components transform
like a Lorentz vector. If it is a second rank Lorentz tensor like the
electromagnetic field tensor it transforms like a second rank Lorentz
tensor.  So given the fields of one observer you can calculate the fields
of the other once you know their Lorentz transformation properties. J.D. Jackson's Classical Electrodynamics describes all of this in the chapters on relativity.
A: Two observers moving relative to each other will not agree on any “same point in space”, but they will agree on the “same event in spacetime”. They will each label it with different coordinates, but they will agree on all of the physics at that event.
Note, since they will have different coordinates, if they try to describe things in terms of their coordinates then those descriptions will differ, but they will be talking about the same thing. The only likely confusion is if they describe things in terms of their coordinates without realizing it.
A: Classically, the Galilean transform is just a skew (shear) transformation: the time axis gets tilted for the moving observer, but both observers agree on what's simultaneous, as well on lengths and on the clock ticking rate:

The observer in the car can keep track of a point that's stationary with respect to the observer on the ground; it's just a linear function of time. This is how they can agree on what they mean by "same place".
Suppose they measure a (scalar) field between spatial coordinates 0 and -1 (the range has no particular significance, it's just that it was more convenient for me to place it on the left side when making the images). Suppose also that the field value is the same everywhere in space, but that it oscillates in time - this is represented by the wavy pattern in the picture.
For a field that transforms as depicted, both observers will measure the same value at what they agree to be the "same place"; they just need to figure out how to transform the spatial coordinates.

But notice that even in this case, what they actually agree on is the values of the field for the same spacetime events (t, x, y, z), it's just that the nature of the transformation means that they don't have to worry about the time component, once they sync their clocks.
Consider now the analogous situation in the setting of special relativity:

The observer in the rocket can do the same trick to establish a stationary point with respect to the astronaut, this time using the Lorentz transformation. However, because their simultaneity hyperplanes are tilted, the two observers no longer agree on what's simultaneous. Also, their clock ticking rates don't match anymore, and they no longer agree on lengths of objects (assuming they don't manipulate their state of motion).
They are literally seeing different times (different events within the same overall process) at different ends of their corresponding measurement ranges. (Note that the event where the astronaut performs the measurement is in the future from the perspective of the spaceship).

So to coordinate their measurements properly, it's not enough for them to sync their clocks once, and then just agree on what's the "same place", they need to figure out what constitutes "same place and time" for every possible (t, x, y, z) combination - i.e. what is the "same event" (an event being a point in spacetime).
Now, as people more knowledgeable than me have pointed out, what values exactly the two observers measure, and how they should reconcile these measurements, depends on the nature of the field itself (i.e. on what exactly the field is describing).
E.g. if the scalar values are sort of "intrinsically" associated with each spacetime point, then it's enough to just figure out what the corresponding events between the two frames are; the values will be the same for same events. But if the field describes something that's affected by this mixing of space and time, then two observers may need to take additional things into account. E.g. if the field describes some sort of density, than the length contraction effect might affect the value. Of course, the field can be of a more complicated nature still.
