It is not clear, but if the dice are an issue, the default is to use the more recent effect
In the PHB rules on "Combining Magical Effects" (PHB, p. 205), the bless spell is given as the example of spell effects not being combined (not 'stacking'):
The effects of different spells add together while the durations of those spells overlap. The effects of the same spell cast multiple times don’t combine, however. Instead, the most potent effect—such as the highest bonus—from those castings applies while their durations overlap. [Errata: Or the most recent effect applies if the castings are equally potent and their durations overlap.]
For example, if two clerics cast bless on the same target, that character gains the spell’s benefit only once; he or she doesn’t get to roll two bonus dice.
This section of the rules is about how multiple castings of the same spell don't combine, but rather, the most potent effect applies. Bless is then offered as the defining example. Bless is supposed to be the example of when the most potent effect applies. This predisposes me to think that we should be trying so see how it could actually work as that example, rather than insist that it is actually an example of when the most potent effect does not apply because all bless spells are equally potent.
The problem is that as the example is written, there is no way to determine which of two bless spells is more potent, due to the line "he or she doesn’t get to roll two bonus dice". If each bless spell offers a d4 bonus, the only way to determine which is more potent would be to roll two dice and compare them.
It could be that the writers really intended that two dice not be rolled. However, one then wonders why they selected bless as the single defining example of the "most potent effect" rule. Or it could be that they actually intended to say something like "he or she doesn’t get to roll two bonus dice and add them both", but poor editing eliminated that crucial phrase.
Operating under the premise that this particular passage might have had poor writing or editing, we can find other parts that do not agree with the rest of the PHB.
For example, the bless spell itself has a duration of "Up to 1 minute" and permits a benefit "Whenever a target makes an attack roll or a saving throw". Thus, the target could potentially benefit from the spell multiple times, even dozens of times, but this passage says "that character gains the spell’s benefit only once."
Furthermore, while it is the character who is receiving the benefit of the spell, it is not "he or she" that is rolling the dice as this passage says; rather, it is the player who rolls.
Considering that the passage as a whole has parts that contradict the spell description, I think it is at least possible that the "doesn't get to roll two bonus dice" line also suffers from similar poor editing. If the spell is, as stated, an example of taking the highest bonus, I think the passage likely intended to say something like:
"For example, if two clerics cast bless on the same target, that character gains the benefit of only one spell in each instance; the player doesn’t get to roll both bonus dice and have them both apply, just one of them in each instance."
Other answers to this question have argued that the "most potent effect" of a spell is entirely contained in the spell description, and that since the bless spell description says it is a d4, all bless spells are equally potent (although they have not so far explained why bless was then chosen for the example of most potent effect).
Consider the repeated prior use of the word "effect". I think a spell's effect is what actually happens to the character. The spell description is the cause; what it does to the character is the effect. In this case, the spell description permits a d4 bonus to the character's next appropriate roll; the actual bonus becomes the effect when it is applied (eg, +3 on a specific roll). In my view, if a character is under two bless spells at the same time, their player should be allowed to roll two different d4 and apply the single "most potent effect" to their next attack or save.
The bless spell is reasonably simple, having a single effect at a time and no save DC: possibly this was why it was chosen as the example. For more complicated spells there is no rules guidance and no clear consensus on how to rule which is most potent. For a character under the influence of multiple Moonbeams, for example, what should determine the most potent effect - the potential HD of damage, the DC of the save, a maximum likelihood of damage that takes into account both HD and DC, the hp of damage actually rolled, or the actual damage that would taken after a save?
The opinions on this issue are abundant and lack resolution (cf):
Fortunately, in cases where the "most potent effect" cannot clearly be determined, there is a fallback: the PHB errata which states "Or the most recent effect applies if the castings are equally potent and their durations overlap." Since the effects of two spells cannot be resolved at the same time, there is unambiguously a more recent effect even when there is not clearly a more potent effect.
So, at tables in which everyone is in agreement that only one die is rolled between the two bless spells, one is rolled. In tables where everyone agrees to roll two dice and apply the better, that is fine. If the table, PC's and DM, cannot come to a consensus, then the more recent effect has the clear precedence.