This depends on the aircraft. However, unless specifically limited to flight from the left seat for a particular make and model (or front/aft in the case of a tandem arrangement), you may fly from either seat.
Note that your pilot certificate doesn't restrict you to left seat operations.
You will find that some aircraft are restricted in terms of equipment or accessories on one side of the aircraft. Many piper twins, for example, don't have brakes on the right side. Some aircraft may utilize flight controls on one side only, by design or by alteration. Instrumentation may be difficult to read due to parralax (reading looks different when viewed from an angle), and some instrumentation may not be readable at all. Further, you may end up working different controls such as power, flaps, gear, etc, with a different hand that you are used to. This can lead to some level of confusion during rushed times; not good during a landing or go around.
As was indicated before, the view is a little different, but this is the least of your worries. It's a very easy transition, and while you can get someone to go with you, it isn't absolutely necessary. You may find that due to insurance requirements, getting someone to "check you out" for the right seat may be difficult when renting. (Some insurance policies will specify that the person in the left seat will be the PIC, and the insurance companies are often the final and bottom line, far beyond the limitations imposed by the FAA). With respect to getting checked out, each single seat airplane I've flown involved a checkout on the wing; I was given a cockpit familiarization before flying, but that was it. First flight is first solo in that airplane, period. So yes, you can do it alone...but consider your limitations, your experience, the airplane, the conditions, the insurance, the owner, and any other applicable factors before you do so.
Yes, it's legal. If your concern is being able to get out after a crash, vs. being able to fully handle the airplane comfortably and safely, examine those priorities once more before making this decision. Good luck!!