Yes, they can attack each other.
The red circle can make melee attacks against the blue square (and vice versa) with no cover.
For burst attacks, the penalty is a bit more situational.
Close burst or blast attacks will not take any penalty, since there is no cover.
Area burst attacks against the blue square may or may not be penalized for cover, depending on the origin square of the attack. If the origin square is on the blue square's side of the door, and there are no other obstructions, there's no cover to penalize the attack.
If the origin square is on the red circle's side of the door, the penalties vary depending on the location. There will either be a -2 for cover, -5 for superior cover, or no attack at all if there is no line of effect.
If the origin square is on the red circle, there is no cover - just like a melee or close attack.
Rules for determining cover are in the Player's Handbook, page 280 or Essentials Rules Compendium, page 219. Rules for determining line of effect are in the Player's Handbook, page 273 or Essentials Rules Compendium, pages 107-108. Pat Ludwig has also quoted some of the important details in his answer.