I think that to ask this question, we would then necessarily have to also ask the question if Manjushri exists? If Tsongkhapa exists? (since Dorje Shugden is an emanation of Manjushri)
Now, Tsongkhapa is an emanation of Manjushri, Chenrezig and Vajrapani. So if we're questioning this, then we also need to question if Chenrezig exists? If Vajrapani exists? And then, if all yidams exist?
Second important question to ask ourselves is what we mean by "exist"? It is important to understand what existence means, in this case. Does he exist according to what WE define as existence? I.e. is there really a red man on a snow lion sitting on a lotus cloud? Or are we talking about the existence of the state of mind / a energy that he represents? In Buddhist teachings, the Buddhas and enlightened mind "exists" in three forms:
- Dharmakaya, which is the ultimate form that these Buddhas exist/abide in which is formless, colourless, odorless etc.
- Sambhokagaya, which is the form that they manifest to us in through visions to Lamas etc. such as as Dorje Shugden, Vajrayogini, Heruka etc.
- Nirmanakaya, which is the form they physically and literally manifest in, such as in Tulkus (Pabongka Rinpoche as Heruka, Gangchen Rinpoche as the Medicine Buddha etc).
That may be a little heavy going on the terminology, sorry!
As to whether we can ascertain their existence with our own direct perception, sight, vision etc we are not yet at the level that we can directly perceive these Buddhas and be sure that they are not just some crazy hallucination! But as explained by Dharma Space and his Lama in the first post on this thread, we can definitely ascertain their existence by inference / inferential logic, such as through seeing and meeting the oracles who take trance of these beings like Dorje Shugden; also by an element of faith that we have in our teachers who can perceive them directly through their clairvoyance and high levels of attainments.
