Parser for gleaning data from twitter
I'd focus on 3 main things:


don't use time.sleep() to wait for elements on a page. With hardcoded time delays there is a tendency to wait more than actually needed most of the time and less than needed sometimes - not reliable at all. Instead, use Explicit Waits with WebDriverWait class and a set of Expected Conditions
remove the overhead of firing up a separate Chrome instance - collect the links into a list a reuse the same WebDriver instance - you should expect improvements in page load times as well 
improve your locators - XPath locators are generally the slowest - use "by id" locators whenever possible; handling class attributes with CSS selectors is more reliable (raw contains() in XPath may generate false positives - it can be a bit better with concat())


In the end, you should have something like this:

from selenium import webdriver
from selenium.webdriver.common.by import By
from selenium.webdriver.support.wait import WebDriverWait
from selenium.webdriver.support import expected_conditions as EC


def browsing_pages(driver):
    driver.get('https://twitter.com/login')
    wait = WebDriverWait(driver, 10)

    driver.find_element_by_css_selector('form.signin input.email-input').send_keys('username')
    driver.find_element_by_css_selector('form.signin input.js-password-field').send_keys('password')
    driver.find_element_by_css_selector('form.signin button[type=submit]').click()

    view_all = wait.until(EC.element_to_be_clickable((By.CSS_SELECTOR, ".view-all a.js-view-all-link")))
    view_all.click()

    # wait for a profile link to become visible
    wait.until(EC.visibility_of_element_located((By.CSS_SELECTOR, ".stream-item-header a.js-user-profile-link")))

    links = [link.get_attribute("href") for link in driver.find_elements_by_css_selector(".stream-item-header a.js-user-profile-link")]
    for link in links:
        scraping_docs(driver, link)


def scraping_docs(driver, item_link):
    driver.get(item_link)

    # gleaning information of each profile holder
    for prof in driver.find_elements_by_css_selector(".route-profile"):
        name = prof.find_element_by_css_selector("h1.ProfileHeaderCard-name a.ProfileHeaderCard-nameLink")
        tweet = prof.find_element_by_css_selector(".ProfileNav-value")

        _, following, follower = prof.find_elements_by_css_selector(".ProfileNav-value")[:3]
        print(name.text, tweet.text, following.text, follower.text)


if __name__ == '__main__':
    driver = webdriver.Chrome()

    try:
        browsing_pages(driver)
    finally:
        driver.quit()


You may also go for a class with the driver instance kept at self.driver, but I think it's pretty much okay to do it this "functional" way considering that you have only two functions. Good read on the subject:


Stop Writing Classes
Start Writing More Classes