In the Binitarian view, who or what is the Holy Spirit? Is it the spirit of the Father, or the Son, or both, or is it a separate entity?
Most binitarians think that the Holy Spirit is, in some sense, God's power. A number of faith statements from binitarian denominations say something to this effect; for instance, here's what the Living Church of God has to say on the matter:
The Holy Spirit is God’s out-flowing power (Luke 1:35). It is the means by which He created and brought into existence the very universe (Psalm 104:30). It is the power by which He works in the minds of human beings made in His image (Genesis 6:3). It is the power by which the lame miraculously walked, the blind saw, the deaf heard, and the dead were raised during the earthly ministry of Jesus Christ (Luke 5:15–17).
Similarly, the United Church of God (Canada) says this:
The traditional Christian view of the Holy Spirit presents it not only as the power of God, but also a third person with the Father and Son in a Trinity. But when the Bible is taken at face value it's clear that it's never presented as a separate person. It's always presented as an attribute or power of God.
So, in short, binitarians (or at least, the major binitarian denominations) teach that the Holy Spirit is the power of God, rather than a separate person. I should clarify that I myself am not a binitarian; I merely hope to be informative.
According to at least one Binitarian group, the Father is sometimes called the Holy Spirit, and the Son is sometimes called the Holy Spirit, and that the Father and the Son have one single divine Spirit. The Holy Spirit is not a separate person.
The reference is their 22 Theses of Binitarianism Thesis 14
We affirm that the Father and the Son are one in spirit. We affirm that together they share one and the same divine spirit. We affirm that the Father, the Holy One, is occasionally called the Holy Spirit. The Son, the Lord Jesus may also be called the Spirit. We affirm that most references to the divine spirit are focused on the Father who is spirit, with some focused on the Son, who is also the spirit of life. We also affirm that their one spirit is holy and is to be reverenced. We deny that the Father and the Son have two separate divine spirits. We deny that the spirit of the Father is someone other than the Father Himself, or that the spirit of the Son is someone other than the Son himself.
We deny that the spirit of God is someone other than the Father, or the Son of God, or that the spirit of the Lord is someone other than God or the Lord. We also deny that words like “the spirit of God,” require a relationship of God with someone else who shares His one divine essential spirit.
And they reference the following scriptures for this position: Mat. 10:20; John 4:24, 14:20; 17:20 – 23; Acts 2:32 – 36; Gal. 4:4 – 6; Rom. 8:9 – 11; 1 Cor. 12:11 - 13, 15:45; Eph. 4:4; 1 John 4:13