A spell's effects don't begin until after the casting is complete
The rules describe the casting of a spell thusly (emphasis added):
A spell is a discrete magical effect, a single shaping of the magical energies that suffuse the multiverse into a specific, limited expression. In casting a spell, a character carefully plucks at the invisible strands of raw magic suffusing the world, pins them in place in a particular pattern, sets them vibrating in a specific way, and then releases them to unleash the desired effect--in most cases, all in the span of seconds.
I've highlighted the important part in bold: a spell doesn't take effect until the casting is complete. In most cases, the casting takes only a few seconds (i.e. 1 Action). But in the case where it takes longer, there is no effect until the entire casting time has elapsed. The fact that the casting time of scrying is the same as its duration is just a coincidence and doesn't change this basic fact of spellcasting. For Scrying, this means you spend 10 minutes (100 consecutive rounds) taking the Cast a Spell action while maintaining concentration the whole time, and then the casting is complete and you begin scrying (if the target fails the save). If you started casting the spell as the clock strikes noon, then your window of scrying lasts from 12:10 to 12:20, assuming you concentrate on the spell for the full duration.
To see why it has to work this way, consider the rules for longer casting times:
Certain spells (including spells cast as rituals) require more time to cast: minutes or even hours. When you cast a spell with a casting time longer than a single action or reaction, you must spend your action each turn casting the spell, and you must maintain your concentration while you do so. If your concentration is broken, the spell fails, but you don't expend a spell slot. If you want to try casting the spell again, you must start over.
Note that the spell slot is not expended until the casting is complete. If the spell could have an effect before then, that would allow you to get the spell effect, then cancel the casting and reuse the spell slot. In the case of scrying, this would allow you to spend 99 rounds scrying on someone for free before canceling the casting at the last round, and then immediately start scrying again 6 seconds later for 99 more rounds, repeating indefinitely. This is clearly not the intent for how scrying is supposed to work.