Sauron, for the most part, is a distant menace in LotR. He doesn’t leave Barad-dûr at all over the course of the trilogy (and yes, he does have a corporeal body and could leave if he wanted to). This chapter is the exception, when Pippin of all people gets to have a little meeting-of-the-minds with the Dark Lord.
The good guys are extremely lucky here. The entire plan of casting the Ring into Mount Doom depends on the notion never occurring to Sauron - any hint that it’s an idea his enemies might even entertain, and with the most minor of efforts Sauron could have rendered Frodo’s quest impossible. Gandalf and Elrond knew and were counting on this, but it’s a flimsy shield and the reason for all the secrecy and deception. Because Sauron was neither stupid nor careless, he just had a blind spot. If the possibility of trying to destroy the Ring was brought to his attention, he wouldn’t have rejected or dismissed it - he would have made extra sure it was impossible, and probably laid a trap to capture the Ring-bearer in the process.
So all in all, having any of the people who knew about the plan communicating in any way with Sauron is a Bad Thing. Someone with a great enough will might be able to resist and challenge him, as Aragorn will later. But poor Pippin? Not a chance. They only managed to escape, as Gandalf points out, because Sauron wanted to question the presumably-imprisoned Hobbit in person. Pippin would have answered any questions Sauron posed.
Overall, this chapter serves as a conclusion of Saruman’s part of the story and a refocus on Sauron. Saruman was a distraction. A powerful distraction, and dealing with him was an absolute necessity, but it served primarily to draw the strength of Rohan away from Minas Tirith where it was needed. Between Pippin’s encounter with Sauron, and the appearance of the Nazgûl west of Anduin, and Gandalf’s abrupt departure with Pippin and his call for them to ride to Gondor with all speed, we get a sudden sharp reminder that the danger isn’t passed, but rather about to get greater.
And so concludes Book I of The Two Towers.
Next post, we finally get back together with Frodo & Sam.