Nothing in the grapple rules indicates that the grappled creature or the grappler change spaces.
The rules for grappling state:
When you want to grab a creature or wrestle with it, you can use the Attack action to make a special melee attack, a grapple. If you're able to make multiple attacks with the Attack action, this attack replaces one of them.
The target of your grapple must be no more than one size larger than you and must be within your reach. Using at least one free hand, you try to seize the target by making a grapple check instead of an attack roll: a Strength (Athletics) check contested by the target's Strength (Athletics) or Dexterity (Acrobatics) check (the target chooses the ability to use). You succeed automatically if the target is incapacitated. If you succeed, you subject the target to the grappled condition. The condition specifies the things that end it, and you can release the target whenever you like (no action required).
The grappled condition says:
- A grappled creature's speed becomes 0, and it can't benefit from any bonus to its speed.
- The condition ends if the grappler is incapacitated (see the condition).
- The condition also ends if an effect removes the grappled creature from the reach of the grappler or grappling effect, such as when a creature is hurled away by the thunderwave spell.
Nothing here indicates that the grappled target changes the space it occupies (unless, of course, the grappler attempts to move the creature somewhere else), and nothing indicates that the grappler changes space. So after a successful grapple, each creature is in the space they were in prior to the grapple attempt. If just grappling (and not dragging) affected positioning, the rules would tell us.