Yes, contact is more common. But, as an example of what I am talking about: After the fight with caster, Rider in Fate/Zero was drained to the point of uselessness. Due to refusal to take from Wavers mana. Acting only on the mana of his own mighty and heroic spirit. And as a result, was so weak he couldn't even take on Physical form, and was even undetectable by Saber due to his presence being as feint and weak as a light breeze. Waver Velvet, his master. Camps outside in safety, eating quite a bit beyond his fill in order to stock up on energy. While Rider drained mana until he can take form once more. Masters can basically throw their mana out into the air, for their servants to take. Im not sure if it is directly given, or if the Servant just takes whats exerted in the air. But they can get it without any physical or present contact, it just leaves the master quite weakened and vulnerable. Especially in Wavers case in that scene since if a battle at all happened, Rider couldn't protect him. Not even as much as raise a single corporal finger.
Moon Hound: Ya mana stats do vary. They change due to the servants reliance on the master for regenerating Mana. Although they do, as Zaer said start with their own reserves. (In Iskanders case it is I believe incredible! Only using Waver for mana, once. One time. Whole series just used his own reserves.) They can not replenish it. The masters mana output is even more important then how much mana the servants start with. Since if the master can not regenerate mana quickly, the servant is at a disadvantage. So a master who can take in a large amount of mana at one time, regenerate it quickly, and send it to their servant just as quickly gives the servant a A++ .
Did that make sense on explaining the Mana stat?