There is no Western Buddhism - Yet
I keep up with a number of blogs and Facebook groups dealing with Buddhism, and I've noticed a lot of talk recently about Western Buddhism. "Western Buddhism is Dying", "I've Fallen out of Love with Western Buddhism", etc. My problem with these is that there is no Western Buddhism - at least not yet. There are lots of people trying to be Japanese, or Tibetan or Thai or this or that; but little to no signs of a Western Buddhism.
I am not saying that these people are playing at Buddhism or not committed to the Dharma. Most of them are devoutly practicing the Dharma in these forms and doing great good in both their own lives and in their communities. One of the finest Dharma teachers I know is practicing in the Tibetan tradition, and she is my ideal of a living bodhisattva. However, what they are practicing is Japanese Buddhism, or Tibetan Buddhism, or Thai Buddhism, etc. They are not practicing Western Buddhism; so I find no validity in criticizing Western Buddhism as it doesn't yet exist.
This is not really surprising when we look at how these other traditions themselves developed. Buddhism had been in China for over 500 years before Bodhidharma "came from the West" to found what eventually became Zen. It was another 700 years before Dogen founded Soto Zen in Japan. In the modern West there was little knowledge of Buddhism at all until the close of the 19th century and few places where Westerners were practicing it prior to the 1960's. This means that compared to the development of Zen, the West has had Buddhist practice for 5% of the time.
There are groups trying to build a truly Western approach to Buddhist practice. The Triratna Buddhist Order, founded in 1967 by Sangharakshita in the UK is probably the largest of these. While drawing from Theravada, Mahayana and Vajrayana teachings they are not associated with any of them. It now has adherents around the world and is probably the most successful of the Western movements. Some people point to the Plum village Community, founded by Thich Nhat Hanh in France, as another Western order is much more appropriately described as a Western expression of Vietnamese Buddhism. The same could be said of KTD - Karma Triyana Dharmachakra - and its attendant KTC centers - except as being a Western expression of Tibetan Buddhism. While both of these last two groups have many Western adherents and teachers they are still based heavily in the traditions and practices of their home countries.
There are some great teachers in the West who teach something approaching a Western Dharma - Joan Halifax Roshi, Bernie Glassman Roshi, Charlotte Joko Beck, Grace Schireson, etc. - yet in their practice they are traditionally Zen - though some more and some less. These would, for me, be the first steps in the development of a truly Western Buddhism - traditional teachers beginning to move their teachings outside the traditions in which they practice; but there needs to be more. Eventually, Western Buddhism needs to develop its own identity - potentially an ecumenical one like Triratna - communities and teachers. There are probably going to be major objections to this development by "traditional" Buddhists since it is eventually going to mean breaking from the traditional structures and hierarchies which have grown up in Asian and East Asian countries over the past 2600 years.
We Westerners, as well, are going to have to be very careful in this development. We cannot reject practices out of hand just because they exist in traditional Buddhism. Many of them exist because they have been found over generations to be skillful and appropriate for the well-being of people. Nor can we embrace anything simply because it's modern or fun. In all cases we should be guided by the Kalama Sutra to make sure the practices we choose, "are skillful; these qualities are blameless; these qualities are praised by the wise; these qualities, when adopted & carried out, lead to welfare & to happiness". This is going to be a long and winding road; but it can be done.