My understanding of relativity is that change in an object slows down the faster it goes, hence the perception of time in a subject will also slow down if he travels a good deal faster than a more stationary subject. But both subjects remain in the same instant of the present. There aren't two presents (Santa is tightfisted).
I don't quite get the point about "here". Different objects are naturally in different locations. Two objects can't occupy the same space.
Palms up, this is a very artificial situation I'm about to describe, but it's tailored to illustrate the point without making the example unreadably long and full of caveats. I'll preface it by pointing out that, in it's most basic form, special relativity only deals with non-accelerating frames of reference. You can 'hack' it (lots of folks would take issue with that term, but it's always struck me as apt) to deal with accelerations, but this is something that general relativity does more explicitly and comprehensively.
We have Al and Bert, two astronauts (they could equally be a hydrogen molecule and a galaxy, they just represent two reference frames) who are in a part of space where they've no measurable reference points but each other. They're moving - coasting - relative to each other with some significant speed. Special relativity's big postulates (which sounds like a euphemism for a boil to me, but I digress) are that everyone has to measure the speed of light as the same (but there's no strict single value for any other measurement) and everyone measures distance and time, from their own positions, the same way (i.e. the laws of physics are the same in each reference frame). Al and Bert both have a clock, a torch and a tape measure of some sort to measure things with.
With no reference point except each other, neither one knows if Al is still and Bert moving, Bert still and Al moving, or both moving with some fraction of their total relative velocity. Since they are coasting, so they feel no forces and stay at constant speed relative to each other, all three scenarios actually appear exactly the same to both of them. Because they have to both see the speed of light as the same value, they both see the
other astronaut's measurements of time and distance being slow and short, by the same amount. Since turning around, and meeting up to compare, would introduce an acceleration they can't do so to see whose clock is actually faster, without leaving the situation our bare-bines special relativity deals with.
If they disagree on whose clock is faster they're going to get two different measurements for each other's speeds, or the speed of any 3rd object we introduce. So they don't agree how long it's been between two events happening, and so they don't agree on where they were relative to each other when they happened.
If we now introduce a third astronaut, Una: She disagrees with both Al and Bert about whose clock is fastest, and about where everyone was at any given time. If we have a given sequence of events - Al flashes his headlights, Una lights a cigar, Bert sneezes - they all see the timing and position of each event differently, and you can easily create scenarios where they even see them in different orders. So everyone has a different take on where they are at any given time, what the time is from any given person's viewpoint, and what order things are happening in.
To be fair: As the entity magically setting up this artificial scenario we're free to just say 'this is how it is' and set the positions, speeds, and times of each astronaut at any given moment. But that only works if we stay outside the scenario, and use 'creator's privilege' to just dictate those measurements. If we have to use light, measuring sticks, and clocks to find out the measurements, the way our astronauts do, and if they can see and measure us the same way, we become just like them and we just have another viewpoint they can disagree with.