The problem of Western science is that it leads to an upside down view of reality. The objects are placed before the subject (consciousness). Can there ever be proof that a world exists outside of consciousness when you would not even know about the world without our awareness of it to begin with? But as Western scientists would have us believe, consciousness came into existence millions or billions of years after the origin of the Universe. The biggest mistake is that they've made all of this conclusions without understanding the origin and nature of consciousness. They have not explained how the brain causes consciousness but they still state that it originates there.Many scientists will tell us that everything is made of matter. If true, then given the properties of matter (having mass and occupies space as a liquid, solid, or gas), then it should be observable. Being observable to the senses would also mean that it's not private or isolated to the subject. But as it stands, consciousness is not observable and it's private (subjective). So it seems one big problem is that scientists who are working under the assumption of materialism, are trying to explain how private and insensible phenomena fits with physical and sensible phenomena. The two are obviously not the same so trying to equate the two is unreasonable. However, the two are connected and the real challenge is showing how.
"What is the nature of consciousness?"
Scientists have not discovered consciousness in its pure form. What they have is how consciousness is expressed within the brain. Consciousness in its pure form is simply awareness. It is not limited to time, space, a brain, a person, to thoughts, to memory, etc. It is simply the witness to existence. Just as in a dream, it can manifest as anything or any person.
"Does this nature pose a problem for science?"
Pure science seeks knowledge and adapts its methods accordingly. Western science is built on a conclusion of materialism so it is not pure science. As such, it views everything as objects. This is why it relies on third-person perspective because it believes all things are objects. It tries to leave out the subject, but it can not. Furthermore, the nature of the subject is, as you say, private and non-physical. This can not be studied with third-person methods, but only with first-person methods.
In contrast, Eastern thinkers have spent thousands of years using a systematic approach to explore reality. They do not make the error of placing the object before the subject. As such, we have reached the conclusion that consciousness is fundamental. To discover the nature and origin of the Universe, one must look within, and our primary tool for that is meditation.
"I regard consciousness as fundamental. I regard matter as derivative from consciousness. We cannot get behind consciousness. Everything that we talk about, everything that we regard as existing, postulates consciousness."
Father of quantum theory, Max Planck