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We live in the era of the most powerful particle colliders ever built, reaching unprecedented centre-of-mass system energies $(\sqrt{s})$ and luminosities. The attention of the high-energy physics community is focused on searches of the New Physics phenomena and measurements related to the rare processes within the Standard Model, including production of the Higgs boson, discovered in 2012. However, there are numerous physics processes of significant contribution to the total cross section, which are not well measured nor described theoretically, still being a subject to studies. Such processes are often recognised as a background in the analyses of rare process and thus their mismodelling leads to large uncertainties of the results of such analyses. Among the aforementioned class of physics processes are diffractive interactions, occurring in the high-energy limit via exchange of the colourless object called the Pomeron. Diffraction is experimentally revealed by presence of the rapidity gap, or gaps, in the topology of the final state. It contributes about 30\% to the total proton-proton cross section at the centre-of-mass system energies achievable at the two proton-proton colliders currently in operation: the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC). The inelastic diffractive processes can further be divided into single, double and central diffraction. The last one proceeds via the Double Pomeron Exchange (DPE), which occurs when each of the interacting beam particles emits a Pomeron and scatters at small angle. The Pomeron s form a neutral state $X$, while the beam particles get excited, dissociate or stay intact. If all particles of the state $X$ are well separated from the beam particles or their remnants, we talk about the central exclusive production (CEP), written as $B_1+B_2 \rightarrow B_1^{(*)}+X+B_2^{(*)}$. CEP in high-energy proton-proton collisions is dominated by the DPE, with additional contributions from photon-Pomeron and photon-photon interactions, the last one being non-diffractive CEP. In this dissertation the study of diffractive CEP in proton-proton collisions with the measurement of the forward-scattered protons is presented. The process is measured in the STAR experiment at RHIC and the ATLAS experiment at the LHC, at $\sqrt{s}=200$~GeV and 13~TeV, respectively. It is the first time when CEP is measured at such high centre-of-mass system energies with the detection of forward-scattered beam particles. This is enabled by dedicated devices housed inside the Roman Pot vessels, which are mounted at the accelerator beam pipe and which allow detectors to closely approach the beamline and to tag particles scattered at low angles. As a result, the exclusivity can be confirmed by the momentum balance of all detected particles, rather then inferred from the double rapidity gap topology, as is done in the experiments without these special forward detectors. The thesis starts with the introductory part providing theoretical basis required to understand and follow the flow thereof. Phenomenological description of diffractive CEP developed in the language of the Regge theory is described and the Monte Carlo (MC) event generators implementing various models of the process are introduced. The following two parts contain details of analyses performed respectively at STAR and ATLAS, including description of the hardware and experimental techniques used during the data taking, in an event reconstruction and in the physics analysis. These techniques are conceptually similar but, despite of the same physics process being analysed, required adjustment for the two independent measurements. It is a direct consequence of completely different experimental conditions at STAR and ATLAS. In the last part, the results obtained in the two experiments are discussed and conclusions are presented. The primary results of the physics analyses are the cross sections for the diffractive CEP of identified charged hadron systems, $p+p \rightarrow p+X+p$, where $X=\pi^+\pi^-, K^+K^-, p\bar{p}$ (STAR), and $X=\pi^+\pi^-,$ $2\pi^+2\pi^-, 3\pi^+3\pi^-, 4\pi^+4\pi^-$ (ATLAS), measured within the fiducial kinematic region corresponding to the geometrical coverage of detection systems used at STAR and ATLAS experiments. The reached experimental precision is several times better than the precision of the measurement of CEP with the forward proton tagging at the highest-so-far $\sqrt{s}$, performed by the AFS and the SFM experiments at the ISR. Measurement of $4\pi^+4\pi^-$ central systems in ATLAS is probably the only measurement of such high central state multiplicity in the CEP process. These fiducial cross sections are compared with the available models of the continuum production, of which none is able to describe the data in all studied production channels. It indicates a significant contribution from resonance production, as well as interference effects between different production mechanisms. Single differential cross sections as a function of the invariant mass of the centrally produced $\pi^+\pi^-$ pairs are extrapolated to the Lorentz-invariant phase space region which allows decomposition into continuum and resonant part, together with an identification of the observed resonances. These are $f_0(980)$ and $f_0(1500)$ (scalar mesons), and $f_2(1270)$ (tensor meson). There is no clear evidence for the production of the vector mesons such as $\rho(770)$. It confirms the dominance of the DPE~mechanism in CEP at studied energies and in the available kinematic range. On the other hand, there are strong pieces of evidence for the production of the~$f_0(500)$, a~resonance around the mass of 1370~MeV - presumably $f_0(1370)$, and a~resonance around the mass of 2.3~GeV. Parameters of resonances extracted with the fit to extrapolated invariant mass cross section are the integrated resonance production cross section and the relative phases between the amplitudes, in a few cases also the resonance mass and the width. The production cross sections are found to strongly change with the azimuthal separation of the forward-scattered protons, hence on the incident angle of the Pomeron s in the laboratory in the plane transverse to the incoming beams. It is observed that the scalar mesons are preferentially produced at low such angles. In this configuration, the relative momentum of interacting Pomeron s is reduced. In connection with the fact, that the simplest chromodynamical representation of the Pomeron is a colour-neutral pair of gluons, enhancement of the production in this configuration may suggest some gluon content in the resonant state, or even the gluon bound state ("glueball"). One of the observed scalar mesons, which resembles the described production enhancement, $f_0(1500)$, is generally considered as a potential lowest-mass glueball. The contributions from the non-resonant production to CEP cross sections were extracted from the data and confronted with the dedicated continuum models. Comparisons provided limits on the range of parameters of these models describing the meson form factors and absorption effects related to the rapidity gap survival probability. In the single differential cross sections as a function of the invariant mass of the centrally produced $2\pi^+2\pi^-$ system, the axial-vector resonance, $f_1(1285)$, was identified and the integrated cross sections were determined for two ranges of azimuthal angle between forward-scattered protons. Within the fiducial phase-space, the $f_1(1285)$ production is found independent from that angle. Detection of the forward-scattered protons enabled reconstruction of the squared four-momentum transfer $t$ and later the exponential fit to double differential cross section in $t_1$ and $t_2$ in the $\pi^+\pi^-$ and $2\pi^+2\pi^-$ channels. The extracted slope parameters, in the case of CEP of $\pi^+\pi^-$ pairs, vary significantly along with the studied range of the invariant mass of the central system, and the azimuthal angle between the scattered beam protons. The results from the presented study are expected to provide strong constraints to parameters of the models of DPE in high-energy particle interactions.