I believe there is discussion of the former (I think I have heard this referred to as a "ring girder") as well as latter methods in AWWA M11, "Steel Pipe -- A Guide for Design and Installation" 4th ed. 2004 (as well as of course ASCE MOP 79,"Steel Penstocks" you may be referring to).  I think whether or not ring girders are necessary or should be used could depend on many factors, including size of pipe, desired basic wall thickness (required for internal pressure), level of vacuum or external pressure, spans, loadings etc.

I encounter pipelines resting on top of concrete headstock for crossing rivers. Some are with ring girders and some without.

How do I check if the ring girders are necessary based on the above factors which influence the need for the ring girders?

I do not typically design rather specialized large diameter steel pipe on support systems; however, I believe in some cases (particularly of some large diameter sizes) there may be more than one way to approach such designs.  e.g. if a certain thickness of steel pipe (let's say chosen as a minimum due to handling/availability, suitable for internal pressure etc.) does not result in excessive deflection, maximum localized stresses, mid-span deflection and will not buckle due to vacuum or other external loading etc. with conventional saddle or roller supports, then there would appear to be no need for the (in that case perhaps some extra) expense of and design time/effort ring girder reinforcement.  There are of course even other types of localized bearing or stress reinforcement such as "wrapper plates" , around all or part (like a "shoe") of the pipe circumference etc. On the other hand, if say the size, chosen thickness for internal pressure/minimum availability, and span/loadings on the system result in too high localized stresses or buckling tendency at the supports etc., it could be argued ring girder or wrapper plate reinforcement at the supports might optimize some designs in engineering economic terms (as opposed to say otherwise thickening the entire pipeline just due to localized stress etc. concern at the supports).  I do however also believe design and fabrication of systems with these localized reinforcements can probably get quite complex in their own right, and the cost of this complexity may need to somehow be factored into the process.  
Hopefully you will get response from some actually experienced in optimizing such designs.

just another thought as well -- a ring girder attached to an external structure could conceivably with adequate design be also be used as a means of lateral and/or axial anchorage as well (like a strap-down mechanism or wall collar welded on a pipeline without same)