I definitely believe that US citizens of other states should fall under Eleventh Amendment restrictions because today's society is full of frivolous lawsuits. Suits against states are a dangerous idea, and also harmful and a huge waste of taxpayers' money. National sovereignty is the bigger picture that should go against individual grievances.
The US Constitution's Eleventh Amendment concerns sovereign immunity of the various states. However, the Founding Fathers envisioned the states as semi-sovereign entities in their own right, and in order for that principle to hold true, they must be allowed to be held accountable interstate suits. Otherwise, there is no accountability for torts over state lines.
Yes, I think that citizens of other states should fall under the Eleventh Amendment jurisdiction restrictions, because the most important thing is making sure that the federal government does not get too much power. In the modern world, interests between people of different states and equality in trials are not as important.
To limit the ability to sue another citizen wherever they reside does not seem to agree with the spirit of the 11th Amendment or the written wording of the amendment itself. It does not extend to individuals seeking lawsuits against other individuals across state lines, but instead prevents non-residents of a state from suing an entire state which protects states from vindictive neighbors. The amendment itself does not limit the individual, but rather protects the state as an entity.