Python3, 149 141141 139 bytes

import time,random
i,x=input(),0;l=len(i)
while x<l:x=int(time.clock());print(i[:x]+"".join(chr(random.randint(32,126))for _ in rangein"a"*(l-x)))

Input from stdin.

Eyes version (159157 bytes):

import time,random
p,i,x=print,input(),0;l=len(i)
while x<l:x=int(time.clock());p(i[:x]+"".join(chr(random.randint(32,126))for _ in rangein"a"*(l-x)),end="\r")
p(i)

Python3, 149149 141 bytes

import time,random
t,p,i,x=time.clock,print,input(),0;l=len(i)
while x<l:x=int(t());p(i[:x]+"".join(chr(random.randint(32,126))for _ in range(l-x)))

import time,random
i,x=input(),0;l=len(i)
while x<l:x=int(time.clock());print(i[:x]+"".join(chr(random.randint(32,126))for _ in range(l-x)))

Input from stdin.

Eyes version (163159 bytes):

import time,random
t,p,i,x=time.clock,print,input(),0;l=len(i)
while x<l:x=int(t());p(i[:x]+"".join(chr(random.randint(32,126))for _ in range(l-x)),end="\r")
p(i)

import time,random
p,i,x=print,input(),0;l=len(i)
while x<l:x=int(time.clock());p(i[:x]+"".join(chr(random.randint(32,126))for _ in range(l-x)),end="\r")
p(i)