How about implementing something like the following. There are a few problems with your program, most notably:

1. The sum is being repeatedly printed for every value.

2. You are simply adding 1 to the sum instead of adding the integer i.

3. You are not returning on your function if your user does not enter a positive integer.

4. You have no if statement for if the integer is greater than 0.

def intpicker():
        x=int(input("Please pick a positive integer"))
        sum=0
        if x >= 0:
            for i in range(1,x):
                sum=sum+i
            print(sum)
        else:
            return intpicker()

This code could be further abbreviated, but for all intents and purposes you should probably just try and understand this implementation as a start.

There are a few fatal flaws in your program. See below:

x=int(input("Please pick a positive integer")) #what if the user inputs "a"
sum=0
for i in range(1,x): # this will not include the number that they typed in
    sum=sum+1 # you are adding 1, instead of the i
    print(sum) 
else:
    x=int(input("Please pick a positive integer")) # your script ends here without ever using the above variable x

This is what I might do:

while True: # enters loop so it keeps asking for a new integer
    sum = 0
    x = input("Please pick an integer (type q to exit) > ")
    if x == "q": # ends program if user enters q
        break
    else:
        # try/except loop to see if what they entered is an integer
        try:
            x = int(x)
        except:
            print "You entered {0}, that is not a positive integer.".format(x)
            continue
        for i in range(1, x+1): # if the user enters 2, this will add 1 and 2, instead of 1.
            sum += i 
        print sum