What would be the dynamics of a double-planet system, similar to Earth / Moon, but with both bodies nearly Earth-sized? Researching a sci-fi story involving a "twin-planet" system. Is such an arrangement (however unlikely) physically possible? And if so, must they be tidally locked with each other, similar to the moon, month-long days, etc?
This is a duplicated question but I think it fits better into astronomy than into physics. I'll try to answer all aspects of this question including the consequences.
First, is it possible? As already pointed out it is, like Pluto and Charon. But is it possible for bigger planets, with the size of Earth? Well we didn't find this planet system yet, but the universe is big. There is no physical law that would forbid this system, it might be unlikely and very rare but it would be possible.
Let's assume somewhere is a planet system with two Earth-sized twins. What would be the dynamics of this systems on the consequences for possible living beings on one of this planets?
One consequence is that the gravity center of this system would be between both planets, if they exactly are the same size (same mass) the gravity center would be in the exact middle of both. That means both planets are rotating around this point (while they are rotating around a sun). This would lead to some effects like in a centrifuge and lead to some weird appearances. The gravity point of our Earth – Moon system is already shifted from the geometrical center point of our earth. And this leads into tides. The tides on our double Earth system would be vast (assuming at least one of this planets is just like our earth and has water on it). Complete areas would be periodically covered with water. Most living beings would probably be amphibians. This of course means the planet would rotate. In the beginning, the planets would rotate but they would slow each other down until they show the same side to each other. Just like our Earth did to the Moon. The Moon continually slows down the rotation of our Earth, but with his minor mass this takes a while. This processes would be much quicker on the double earth system. But even this would take some time. So it would be a matter of time when you look at this system. If it's a young system they would still rotate. If it's an older system they would show the same side each other.
The last point would fix the tides problem, because the water level would be stationary. But this would lead into new problems. The day length would be as long as they need to rotated around their common gravity center. I'll be honest. I am not sure how long this would be. For a stable double Earth system the distance between both planets would be further then the Earth-Moon distance. Let's just assume a day in that system would take 2 Earth months for simplification. This would mean on daylight it would be very hot and at night it would be very cold. One side of the planet would be heat up for hundreds of hours (our earth hours) and the other half would cool down for the same time. Temperatures over a period would jump up and down in big steps. There is no living being known on earth that can handle both extreme hot and extreme cold in such a short period of time. Also this would lead into very heavy and constant storms in the area where the sun sets or rises, making living there even harder.
Also your weight (stepping on the bath scale) would vary depend on where on the planet you are. If you on the side facing the other planet you would be heavier than on the other side of the planet.
Another point that would be possible, but I will point out I am not completely sure about this. The planets would struggle keeping their atmosphere for a long period of time because of the rotation. The side facing away from each other would have thicker atmosphere than on the sides (see my sketch).

(source: directupload.net)
At this point the gravitational pull of the planet is also lower then everywhere else. This could help the molecules to escape the Earth, in a long period of time.