I hunted deer over hounds for years. Never saw a "panicked, frightened" deer EVER. I only saw deer loping ahead of dogs they know they can easily out run.
Again, that is my "ignorance" about hounds and deer. I can see beagles being more of a PITA for a deer than a real concern, especially in areas where people regularly run beagles for rabbits and such. There is a difference in terrain as well; in the West many dogs can keep up with deer and can often run them to exhaustion due to the relatively open forests, where as in heavy brush and timber with lots of windfall, etc. deer can easily outrun dogs if they put their mind to it.
Having said that, fast hounds like walkers would be able to push deer pretty fast in good conditions, and as another poster stated they can move deer fast enough to prevent them from thinking too much. That may not be "panicked", but it is definitely "stressed".
On a side note, an observation I have made re: deer and dogs is that where there are stray dogs you will not find deer, period. My work takes me to many reserves throughout AB, and where there is no dog control program you will almost never see deer (or tracks) or elk, or moose. Where dogs are kept under control there is a lot more deer activity. Deer do not like dogs.
ETA: an earlier poster stated they train their deer hounds year round. How does one do this considering the regs and consequences?
"Power of conservation officer
(4) A conservation officer may kill a dog without incurring any liability if,
(a) the dog is running at large in an area prescribed for the purpose of subsection (2) during the open season for white-tailed deer, moose, elk or black bear; or
(b) the dog is chasing white-tailed deer, moose, elk or black bear during the closed season for that species in an area usually inhabited by that species. 1997, c. 41, s. 25.
Dog training during closed season
26. A person shall not use a dog to chase game mammals or game birds during the closed season for the purpose of teaching the dog hunting skills or testing the dog’s hunting skills unless the person has the authorization of the Minister. 1997, c. 41, s. 26."