Some comments on your code. Overall you are using a lot of unnecessary variables.

    another_word = " "
    sentence1 = " "
    new_list = []
    another = " "

You really don't need these variables declared outside of your functions, they add no value and just pollute the namespace.

    def trans_one_word (i):

PEP8 doesn't recommend a ` ` before the args. And you should consider having more meaningful arg names.

        global another_word

Unnecessary global, you are returning this variable from the function so don't need it to be global

        pyg = 'ay'

        original = (i)

Parens are unnecessary and you could just call the arg `original` as you don't need this assignment.

        if len(original) > 0 and original.isalpha():
            word = original.lower()
            first = word[0]
            new_word = word + first + pyg
            another_word = new_word[1:len(new_word)]

You don't need `len(new_word)` on a slice an empty slice implies to the end of the word, e.g. `new_word[1:]`

            first = word[0]
            new_word = word + first + pyg
            another_word = new_word[1:len(new_word)]
            return another_word

This whole block can be simplified to `return word[1:] + word[0] + pyg`

        else:
            print('empty')

You don't need the `else:` clause because the previous block returned. You should also return the original word.

    def input_state():

        sentence = input("enter a word to be translated: ")

Assume you meant to ask for a words or a sentence given the variable name:

        list = sentence.split()

`list` is a python type, so shouldn't be used for a variable name.

        for i in list:
            trans_one_word(i)
            new_list.append(another_word)
        s = " "

You should look into list comprehensions and generators as this can be significantly simplified, e.g. `print(" ".join(trans_one_word(i) for i in sentence.split()))`

        print (s.join( new_list ))

It's probably better to return this result and let the caller `print` it.

    input_state()

    another = input("Would you like to enter another sentence to be     translated? \n Y/N")

    if (another == 'y') or (another == 'Y'):
        input_state()

This condition can be replaced with `if another in 'yY':`
You should really put this all into a loop so with an exit clause.

Simplifying the code:

    def trans_one_word(word):
        pyg = 'ay'

        if word and word.isalpha():
            word = word.lower()
            return word[1:] + word[0] + pyg
        return word

    def input_state():
        sentence = input("enter a sentence to be translated: ")
        return " ".join(trans_one_word(word) for word in sentence.split())

    while True:
        print(input_state())
        another = input("Would you like to enter another sentence to be translated?\nY/N")
        if another not in 'yY':
            break