Проверьте, анимирован ли div, и узнайте положение div

var $width = $("body").width() / 2;;
var $animatedDiv = $(".hero-container");
$("#sign-in").click(function() {
  $animatedDiv = $(".hero-container").animate({
    "left": $width
  }, "slow");

  $("#welcome").html("Sign In");
  $(".form").show();
});

$("#sign-up").click(function() {
  // if the div is on the right, then move it to the left
  if ($animatedDiv.is(':animated')) {
    $animatedDiv = $(".hero-container").animate({
      "right": $width
    }, "slow");
  }
});

<script src = "https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
<section class = "welcome">
  <div class = "hero-container">
    <h1 id = "welcome">Welcome!</h1>
    <p>Welcome back</p>
    <button id = "sign-in">Sign in</button>
    <button id = "sign-up">Sign up</button>
  </div>
  <div class = "form">
    <form class = "sign-in">
      <h1 id = "signIn-heading">Sign In</h1>
      <input type = "text" name = "email" placeholder = "Email">
      <input type = "text" name = "pass" placeholder = "Password">
    </form>
  </div>
</section>