Chenkel said:
I've heard this is reciprocal and if we situate a clock at either reference frame we will measure the clock in the other reference frame to have a longer period than the clock at "rest."
As explained by others before, one can understand the symmetry of time dilation only in connection with relativity of simultaneity.
So let's say we have two clocks A and B at the same position, both indicating 0 at the start.
In the rest frame S1 of A we have another clock C synchronous with A.
In the rest frame S2 of B we have another clock D synchronous with B.
In order to simplify matters, we chose another frame S3 in which S1 (together with A and C) is moving with +v, while S2 (together with B and D) is moving with -v. This has the advantage that all four clocks have the same time dilation as seen in S3, and we furthermore choose v so that the Lorentz factor becomes 2 in S3.
We also observe that when A and C are synchronous in S1, they are not synchronous in S3 due to relativity of simultaneity, but C is
ahead of A by 5 seconds. Likewise, when B and D are synchronous in S2 they are not synchronous in S3 due to relativity of simultaneity, but D is
ahead of B by 5 seconds. This is described in the first part of the following image.
In the second part of the image we see the result when S1 and S2 continue their motion for 10 seconds (S3 time) until the clocks coincide. Since the time dilation factor is 2 and all clocks have same speed, all of the clocks will advance by 5 seconds.
We read off from the second image part: B is retarded with respect to C. However, since C is synchronous with A in S1, the observer in S1 concludes that
B is retarded with respect to A.
We also read off from the second image part: A is retarded with respect to D. However, since D is synchronous with B in S2, the observer in S2 concludes that
A is retarded with respect to B.
Summarizing:
B is retarded with respect to A in S1, while A is retarded with respect to B in S2. Time dilation is symmetric.