Do lapped joists need to extend over the beam the same distance from each side?

As long as both joist go over the beam your are fine. Remember if it bothers you you could always cut one of the joist so it was equidistant over the joist but its total unnecessary.

IMO, theoretically, a well-jointed splice is as good as the joist without a splice. However, the stresses in the wood and fasteners will complicate the issue due to offsetting the splice on the support beam, at which the internal stresses (shear and moment) of the joists are very high, and each joist will have different internal stresses that are difficult to evaluate and may cause problems in the long run. The graph below shows the clamping forces required to close the gaps due to deflection of the joists at the splice, and the shear and moment of a continuous joist over intermediate support (beam).

I suggest either keeping the splice center on the beam or shift the splice away from the beam. Otherwise, pay extra attention to your splicing details.

At this juncture, you shall ask yourself "do I need to splice the joists to make it continuous or not?". If it is unnecessary, you shall trim the joists to the length required for a simple bearing connection, thus eliminating the problem of overhanging that will raise higher over the support and pushing against the floor in contact. Since this (offset splice) is not a common problem addressed by most trade publications, therefore, it pays for consulting an engineer or experienced wood structure designer if you intend to proceed but having doubt.

Your detail will not threaten the integrity of your structure, it may not fail immediately. Rather, potentially it would affect the long-term performance of the structure by building a defect into it. A forum like this is a good place to gather ideas, but you have to make the right call for your own good. Good luck.