Freedom of movement for workers is one of the fundamental building stones of the EU and EU citizens don't need to get prior authorization to work elsewhere in the EU.
In practice, some countries used to (and perhaps still do) deliver some form of residence “permit” but unlike non-EU citizens, you are entitled to get one (i.e. formally the permit just documents your pre-existing treaty rights, it's not up to the authorities of the country to give you one or not). In my experience, it's a formality you complete after starting to work, nothing much happens if you don't get one.
For example in Ireland I showed up a couple of times at the local police station where apparently only one garda knew the procedure and he was never there; his colleagues just told me not to bother. In the Netherlands, I got one from the town administration where I was asked if I want one or not and told it's mostly useful to simplify things like getting welfare benefits if I lost my job, etc. (I don't have personal experience with that but I believe you only qualify for welfare after living several years in the country, you can't go “benefits-shopping” and move to a country just for that).
All of this is not yet true for citizens of Romania, Bulgaria and Croatia who still suffer from restrictions in certain countries for up to 7 years after their country joined the EU under transitory rules and do sometimes need to go through a lot of red tape to get a work permit.
Of course, if you want to practice a regulated profession (starting with lawyers and medical doctors but in some countries the list is really long), you need to get your qualifications recognized. Getting this type of things working as smoothly as possible is a big part of what the EU does but that's unlikely to be an issue for short-term jobs. In many countries, you also need to get a tax/health insurance number that locals would have had since their birth and that employers will require.
In some countries, I suspect working legally would also mean that you need to register as a resident with the local administration irrespective of the time you intend to stay (in Germany, Switzerland, the Netherlands, residents have to officially register any change of address within days of moving – it feels kind of creepy to me but it's very easy to do in practice). So there can still be some red tape involved even if you don't need a permit per se.
Furthermore, a small number of defense, health or policy-related positions can be closed to citizens from other EU countries (but certainly not all public sector jobs). Info on all this on the official EU website is very good.
PS: Immigration is mostly off-topic on this site but then again we do have questions about working in Georgia or Argentina and, if you are a citizen from one of the member states, the EU rules obviously also allow you to work for a short time while traveling around Europe.