That is not correct, unless the second company is different from the first.
As for what happens to holdouts that e.g. do not think the price is high enough, their situation is explained by: http://www.investopedia.com/ask/answers/06/rejecttenderoffer...
The part that is usually not explained is that there are state-by-state, and company-by-company, mechanisms to compel the left-over minority of shareholders who do not exchange their shares to be "squeezed-out" and paid the same cash value everyone else got. After this all of the shares are held by the private owner and the security is delisted/unregistered from the public markets
Typical, but not the only scenarios are a management buy-out and a private entity offering to buy all stock. To make this work, it is usually necessary to offer a premium price.
How do you take something private? What does it mean?