Essential oils are not greasy. They are not the same as cooking oils or massage oils.
You do need to mix the essential oil with a carrier, because it's so concentrated that it takes a very small amount to achieve the desired effect and you want to distribute the oil evenly around the area. Also, some essential oils are considered too potent to be used full strength on the skin and need to be diluted in a carrier oil first to prevent reactions. For the scalp, no more than 1 or 2 drops would be enough of the essential oil, diluted in probably no more than an ounce or two of carrier oil, depending on how much hair the person has.