I have an idea for a DS-like world setting mostly set after the year 4200. The Earth was destroyed around 2500 and humanity moved from earth to an asteroid best and built a giant dyson sphere to live on that has 6 holes in it- North, south and equator. Its radius is 1 AU (average of Earth's orbit) and it's 12500 km thick (roughly Earth's diameter). I have calculated its surface area, volume of its 4 oceans, and a rough population estimate. I'm not sure about the physics aspects of something like this, I want the gravity to make sense and be similar to Earth's on the inside and outside but everything I've found says it wouldn't have internal gravity and ridiculous gravity on the outside. Could it have gravity if it was big/dense enough?
So, I did some calculations and this is what I came up with:
If we assume both structures to be perfectly spherical ( earth included) and both bodies to have the same gravitational attraction ( as you suggested), then with the Earth's mass of 5.972*10^24 kg, for your structure to have the same mass (and therefore, same gravitational attraction), it would have to have an average density of 1.698*10^-6 kg/m^3 in order to have an earth-like mass, with its volume of 3.5151*10^30 m^3
This implies either one of two things:
Humanity has either discovered some material with a super low density or they have some sort of gravity altering device to simulate an earthlike gravity. In today's tech, this is not possible. Keep in mind, the gravity will also vary significantly at different points since you have mentioned that your structure is 12500 km thick.
Since this will most likely require handwavium to make it work, I propose Cavorite from H.G. Wells' "First Men in the Moon". While this material violates physics as we understand it currently, this might help mitigate the gravitational effects of such a massive object especially if it's made of something that doesn't possess a super low density.
Something a little more logical and realistic perhaps is to have your people live high above the surface of the sphere where the gravity will be weaker.
A third option, while still difficult to pull off is to use levitation currents. Superconductors have been known to levitate objects so this may aid in maintaining an earthlike gravity.
Or finally, you could use Larry Niven's gravity generators if you choose to make your sphere out of some super light weight material.