Gathers pair of names (first + last) and separate them into two files
The functionality of your program seems slightly weird. You read multiple files, and each one gets split, but in doing so you (explicitly) delete the old data, is this right?

One thing I notice is this piece of code here:

open("last_names.txt", "w").close() #cleaning the last_names.txt file
open("first_names.txt", "w").close() #cleaning the first_names.txt file
lastNamesFile = open("last_names.txt", "a+", encoding="utf-8-sig")
firstNamesFile = open("first_names.txt", "a+", encoding="utf-8-sig")


This could be replaced by: 

lastNamesFile = open("last_names.txt", "w", encoding="utf-8-sig")
firstNamesFile = open("first_names.txt", "w", encoding="utf-8-sig")


I'd also move it to the start of your function:

with open(fileName, "r", encoding="utf-8-sig") as namesFromFile, 
     open("last_names.txt", "w", encoding="utf-8-sig") as lastNamesFile,
     open("first_names.txt", "w", encoding="utf-8-sig") as firstNamesFile:


So you can remove the close statements at the end. I'm not sure what the encoding= does, but it might not be necessary.

You can also reduce this:

    for names in namesFromFile:
        allNamesFromFile.append(tuple(names.replace("\n", "").split(" ")))

    for namesTuples in allNamesFromFile:
        try:
            lastNamesFile.write(namesTuples[1] + "\n")
        except:
            lastNamesFile.write("n/a\n")
        try:
            firstNamesFile.write(namesTuples[0] + "\n")
        except:
            firstNamesFile.write("n/a\n")


To this:

for names in namesFromFile:
    names_split = tuple(names.replace("\n", "").split(" "))
    if len(names_split) &gt;= 1:
        firstNamesFile.write(names_split[0] + "\n")
    else:
        firstNamesFile.write("n/a\n")

    if len(names_split) &gt;= 2:
        lastNamesFile.write(namesTuples[1] + "\n")
    else:
        lastNamesFile.write("n/a\n")


Allowing a for loop less and two less try ... except blocks as well as a global list which can now be left out:

"Global Variables:"
allNamesFromFile = []


I don't think this is necessary:

fileName = str(input("Please provide the file name with its extension: "))


Instead use:

fileName = input("Please provide the file name with its extension: ")


I'd also look at this part:

try:
    names_separator(fileName)
    print("operation completed")
    break
except:
    print("You have provided an incorrect file name...")
    continue


You are catching a lot of exceptions, try to be more precise. Instead try: 

except FileNotFoundError:


Lastly, you might want to check naming conventions, variables should be in snake_case and classes in CamelCase.

So, to summarize:


check if your program does what you want it to do
open files using with
limit the amount of global variables to 0
limit the amount of try ... except blocks
be explicit in the exceptions you're catching
check pep8 style guide for naming of variables