Electronic – On a Rigol DS1052E oscilloscope, what does the word “Local” mean

Most oscilloscopes have a safety ground. This connects to the earth ground on the plug and also to the case. The thinking is if you wire it up wrong worst case you trip the GFI/RCD instead of shocking yourself or damaging the instrument.

As an oscilloscope is usually a bench instrument it is not isolated from the mains. You can use an isolation transformer, but this is not recommended, as it removes this protection leaving the earth ground floating.

Tektronix sells an oscilloscope which has fully isolated inputs, I don't know of any other models. It is significantly more expensive to add high speed isolation circuitry. Usually it means the whole analog and acquisition side must be isolated.

The ground should always be connected to your circuit's ground or you will get strange behaviour. Trust me. I've spent hours looking at a trace wondering where the noise was coming from, just to find I haven't properly connected the ground.

The simple answer is no, it won't be anywhere near enough.

The reason:

Digital Storage Oscilloscopes (DSO) are limited bandwidth wise by two things: the sampling rate, and the analogue bandwidth. Generally one dictates the other depending on the desired operation of the scope.
For sampling, Nyquist tells us that we cannot detect signals unambiguously above half of the sample rate (i.e. we get aliasing). However, for a scope we want to do more than just detect, we want to see the waveform, so generally a bandwidth of around 1/10th of the sampling rate is used. Also, since a filter starting close to the Nyquist frequency that (e.g. >400MHz) would roll off enough to prevent aliasing is much harder and more expensive to design than a simple one or two pole filter rolling off from further back.
So the analogue bandwidth is usually dictated by these factors, and of course it doesn't make sense to waste money on bandwidth you are not going to use. An exception to this might be if aliasing was desired (i.e undersampling, often used for RF applications) or if the scope is capable of equivalent time sampling (ETS). If most of the stuff you want to observe is repetitive, then ETS may be of use to you - make sure the scope has the required analogue bandwidth as well though.

Some scope product pages are a bit misleading with specs, making it hard for a new user to figure out what is what.
Generally though, a decent scope will advertise the usable bandwidth along with the model number, like for your Rigol, which is advertised as "Rigol DS1052E 50MHz Digital Oscilloscope".
So the bandwidth is 50MHz, and signals above this be be attenuated, more as you get higher in frequency.
Relevant to the above discussion about sampling rate and analogue bandwidth, note that there is no physical difference between the DS1052E 50MHz, and the DS1102E 100MHz. They both have the same sampling rate and front end, but the bandwidth is intentionally limited for the DS1052E using a varicap that can be set in the firmware. It can be hacked to 100MHz, or at least the old model could) by setting the firmware accordingly (using RS232 IIRC)

So for real time observation of signals up to a GHz, you are talking at least 5GHz sampling rate, a front end to match, and a pretty expensive scope (>£5k at least)