Praying to a Buddhist deity for healing of someone How do Buddhists look to deities such as Green Tara or the Medicine Buddha by saying prayers or mantras for another person? How do the deity and the divine operate or function for this purpose? I believe that there can be divine intervention of a divine source. How is it with these deities? When I send healing to another person through reciting a mantra to the Medicine Buddha for example? How does the thought get through? How is the Medicine Buddha or Tara a vehicle for my prayers? 

The sutras speak about the benefits of reciting Medicine Buddha mantras for oneself. What does it do in praying for someone else? Simply, does the deity have powers to heal that person in a divine way? Like through God or calling for the archangels to assist?
There are many misconceptions regarding mantrayana.

In essence, the ultimate nature of the deity is bliss and emptiness, just like the ultimate nature of our own minds.

When we invoke the deity via visualization and/or recitation or chanting of the mantra, we are invoking that aspect of our own consciousness, our Tara-ness or our Medicine-Buddha-ness.

The Buddha Shakyamuni did teach that treasuring someone while in positive states of mind or sending beneficent prayers and thoughts would benefit another being, certainly in the "realm beyond" (afterlife, afterlives) and in the present there can certainly be effects.

It is important to remember that Tara is Prajnaparamita, she is not a self-existent deity, the same for all the deities of the Buddhist pantheon.  Some are called deities to mean deva or god-like rebirth, but in the tradition you are referring to deity means something special: it means that long ago a being aspired to be of benefit to countless sentient beings and through the strength of such aspirations and the diligence and patience of practice one was able to reach a state that helped reflect the fundamental goodness of all beings who practice their mantra/visualization with joy and faith, or devotion.

So what happens when you are invoking green Tara or Medicine Buddha to help heal a friend?  Well, the attitude you have is the most important part.  You are using enlightened Buddha forms as a fulcrum for your own praxis, to adjust your mind and heart attitude to embrace their full healing potential.  This will certainly result in a positive effect, and the waves of which may reach your friend rapidly through causation and intensity, having an ameliorative effect.  At the very least, your mind will become more adjusted to these divine states of being (please see the four unlimited or immeasurable abodes) which is of immense benefit to yourself and to others.

In Buddhism, attitude is just about everything.  Karma is intention, so cultivating the intention to heal and relieve suffering (kindness, bodhichitta) is excellent.

There are many methods, but sometimes it is easiest and fastest for us to relate to our own primordial purity through the practice of such sophisticated "spiritual technologies" such as deity and mantra practice.  There are pure aspects of our consciousness, just remember that their ultimate nature is no different from the nature of mind itself.