No, most priests can’t.
Most priest, however, develop a sort of professional sense about the people they’re dealing with after they’ve been doing their job for a while. In the same way that a mechanic can hear the way an engine sounds and sometimes diagnose the problem with no more information, a priest can sometimes give someone a once-over and give an educated, pastoral guess at what’s bothering them.
I’ll share a brief story to exemplify this.
Once, at one of our Parish Council Meetings, there was talk of changing hours for the Sacrament of Reconciliation. One woman went off about how Reconciliation wasn’t important anymore, and it could be done less frequently, if at all.
During this, the elderly pastor was scribbling on a legal pad he had. Once the woman had finished her tirade, he handed the legal pad to her with the simple words “These are your sins, no?”. The glanced down at the pad, turned pale, and didn’t talk for the rest of the meeting.
After the pastor about how he did that afterwards, his only words were “Once you’ve been doing this for a while, you kind of get a feel for what’s troubling who,”.