Among the answers, I would like to add the practical sense and possibly the sense in which the OP is seeking to understand how the Holy Spirit groans.
Yes the Holy Spirit can speak, pray, groan on His own, without the need of a human body to inhabit through which He can express Himself.
But given that He was sent to live inside of the believer, the Spirit can share the biological machine with the believer. Therefore when the Spirit is using the body, two spirits are experiencing the same event from two different perspectives.
So you have the spirit of man or as Paul would have it, the inner man, living inside the outer man or the carnal man, or the flesh or the body or the biological machine housing. And you can have the Holy Spirit living inside too. (A demonized person will have any number of demon spirits also, another subject, another discussion, for another time).
“Or do you not know that your body is a temple of the Holy Spirit within you, whom you have from God? You are not your own,”
1 Corinthians 6:19
As such the Holy Spirit can opt to take charge while a believer is praying (especially if He is invited and asked to do so) and He will groan using your very body. It’s a shared experience.
You can feel the whole experience, you can if you want, stop it (I don’t recommend it) and it’s felt on all levels, physically (audibly, through the physical senses for example i.e. tears, deep sighing requiring your mouth movement, eye movement) and it’s felt emotionally too, as well as your mind is engaged but often times is fruitless because you’re not sure why you’re groaning, for what reason, you just know that the Spirit is helping you in your praying and you give Him permission to continue.
The Spirit’s groaning are without words, but every time I’ve experienced them, they were extremely significant prayers for very troubling circumstances that would follow in the future. Things I could never have known to pray for and each time very troubling experiences. Every time I credit being able to come through because of those specific groaning prayers. The Spirit actually reminds me and let’s me know, this is what I was groaning about on your behalf back then, and each time I definitely needed it.
You cannot manufacture a genuine groan of the Spirit because it has to be His groaning through your body. If you’re groaning through your body, then it’s not the Spirit doing the groaning, it’s your own groaning. These Spirit groans are special and for me personally they are rare for exceptional circumstances.
“For no prophecy was ever produced by the will of man, but men spoke from God as they were carried along by the Holy Spirit.”
2 Peter 1:21
The men spoke but it was actually the Holy Spirit talking through them. It was not man’s will, it was produced by the Holy Spirit, using the man who gave himself over to be used by God.
This phenomenon is not natural, it is above the natural. Take Samson, His strength required the Spirit to act on Samson’s body
“Then Samson called to the Lord and said, “O Lord God, please remember me and please strengthen me only this once, O God, that I may be avenged on the Philistines for my two eyes.””
What did you ask for exactly? The Spirit to use his body
“Then the Spirit of the Lord rushed upon him, and although he had nothing in his hand, he tore the lion in pieces as one tears a young goat. But he did not tell his father or his mother what he had done.”
If the Spirit of God can tear a lion in two, crash a whole building using the body of a human and Samson experienced the move of the Spirit and shared in the events, The Holy Spirit most certainly can groan using the body of a believer.
Reading over this verse it occurred to me that in the same chapter the Holy Spirit prompts the believer to speak.
“For you did not receive the spirit of slavery to fall back into fear, but you have received the Spirit of adoption as sons, by whom we cry, “Abba! Father!””
We do the crying, we say the words “Abba! Father!” But it’s actually the spirit. In other words for someone on the exterior unless they were given a spirit of discernment, they would conclude that you are crying and saying Abba Father. But Apostle Paul, inspired by the Spirit makes the distinction that this is done by the Spirit.
In like manner, from the exterior a person groaning by the Spirit would appear to be groaning of their own accord, however it’s really the Spirit groaning through them. It’s a very holy/sacred moment, where the believer is merely the ‘vessel’, through which the Spirit is choosing to manifest.