NO. You do not need a partner to start learning how to dance salsa. Nearly all beginner group classes accommodate students with no partners. There are ladies that come by themselves, there are guys that come by themselves. During class we create a circle and we rotate from one person to the next. It's fun, sociable, and you will practice your dance moves with different people.
For private dance lessons, if you do not have a dance partner, you will be dancing with the instructor or assistant instructor if applicable. Remember that the price for private lessons is the same if you come alone or with a friend.
In case you do have a partner, please remember that there are some advantages and disadvantages to dancing with the same person consistently. Advantages of having a partner is that you have someone to practice with on a regular basis, which is great because a lot of practice is very important. However, practicing with only one partner should be avoided, since your lead/follow will adjust to the partner's mistakes or you will learn to automatically anticipate their moves from sheer habit. So as soon as you start dancing with someone else you may encounter problems. While developing the leading and following skills, it is best to dance with a lot of different partners so you don’t rely on your partner to know your moves and compensate for your mistakes.