This is obviously a controversial question, and there's a fair degree of ongoing historical debate about exactly how to characterise and compare political violence and repression on each side. I'll give a slightly bare-bones answer here, as I'm travelling, but can try and at least clarify where there is certainty and uncertainty here.

First of all, there was definitely violence on both sides. This is perhaps an inevitable outcome of a civil war along political lines - each side faced the reality that within their own territory, there was not going to be a homogenous population of supporters, but rather plenty of people who'd much be fighting for the other side. These real and perceived enemies within were a source of a great deal of paranoia, perhaps particularly on the Republican side, as the entire conflict stemmed from betrayal in the first place, and they were soon on the defensive in the war itself. On the Nationalist side, the whole point of the conflict for many of its leaders was the purification of Spain from the evils of Marxism, freemasonry and atheism (or all of the above), and they had no qualms about using violence to achieve this. So, circumstances, and the worldview of the key participants, leant themselves to widespread political violence.

In the Republican Zone, the most famous such violence was directed against the clergy. Anti-clerical violence was a traditional feature of Spanish politics, but it occured on an unprecedented scale in the early weeks and months of the civil war. More priests likely died in Spain than in Russia during the Bolshevik Revolution and civil war. In some regions such as Catalonia, around a third of the pre-war clergy were killed. This widely-publicised violence - which, as indicated by Beevor, was further inflated and sensationalised by the foreign press - was a huge blow to the international legitimacy of the Republic. In this sense, Beevor is correct about the added hype, but it is worth noting that the violence itself was very real. Historians who are particularly sympathetic to the Republic (I'm thinking particularly of Paul Preston's recent The Spanish Holocaust here) tend to blame this violence chiefly on Spanish anarchists, who were indeed the most outspoken about embracing the need for revolutionary violence as a necessary response to the situation. However, I don't think this view is fully sustainable in light of new research over the past decade or so, which has highlighted the role of socialists, communists and others in initiating and carrying out political violence in these early months. There's an important degree of continuity between the perpetrators and organisers or Republican political violence in those early days, and the later Republican bureaucracy, police and intelligence services - those involved in reovlutionary tribunals, for instance, tend to go on to have wartime careers in the justice system. Indeed, participation in these events might even be taken as a badge of loyalty - they had proven their dedication to the cause, and could therefore be trusted participants in the war effort. These kind of continuities I think make it difficult to argue that it's easy to isolate Republican leadership and institutions from what happened.

It's worth noting here that it was these first few months of the war that saw the worst waves of repression and violence on both sides, until the situation stabilised somewhat and the two sides began to feel somewhat more secure (though paranoia about spies and enemies within never faded). However, what is notable is that the violence on the Nationalist side never really stopped, and reached a new crescendo in the final months of the war. This reflects the dynamic of the war itself - conquering territory meant needing to purge it, and it was the Nationalists who did most of the conquering over the course of the war. The final collapse of the Rpeublic, in turn, unleashed new waves of repression against those who had fought the hardest against Franco, including not only mass executions, but a huge system of prisons, concentration and labour camps designed to categorise and punish their enemies. This is one reason why just about everyone agrees - even those who are skeptical of the Republic - that many more (by a margin of tens of thousands) were killed by the Nationalists.

I don't particularly like the comparison with Stalin's purges, personally. For one, the growth of a new security state (with secret police and so on) tended to focus rather than broaden political violence in the Republican Zone - most of the worst violence was a relatively spontaneous early reaction to the emergence of an open political enemy, rather than against largely imagined enemies within during peacetime. By this I don't mean that the violence wasn't deliberate or was justified, only that the particular dynamics of Stalin's purges don't seem to have been replicated beyond a very superficial level. Moreover, the old assumption that the Republican Secret Police were run by the NKVD seems also to be false - while communists played a role, and had their own power bases (especially in Valencia), other political groups maintained their own influence.

In terms of further reading about political violence during the civil war and its aftermath, I'd look into the work of Julius Ruiz in particular, perhaps alongside that of Paul Preston, who is more sympathetic to the Republican position.