Eusebius of Caesarea was, as far as I know, the first Church historian to comment on the history of the New Testament books we call the "Gospels". Eusebius wrote in the late 3rd and early 4th centuries.
According to Eusebius, the contents of the Gospel of Mark are attributed to the Apostle Peter, of whom Mark (or "John Mark") was a disciple. The 2nd century Church Father, Justin Martyr, refers to the Memoirs (ἀπομνημονεύματα) of Peter in his Dialog with Trypho the Jew. Irenaeus, Origen, and Tertullian confirm the tradition that Mark's Gospel ultimately came from Peter.
Matthew and John were Apostles, so they, of course, were eyewitnesses. Luke was not one of the twelve apostles, but he was believed to have been one of the Seventy Apostles that he himself described in his Gospel. I do not believe that the authorship of these three Gospels was held by the early Church to be from anyone other than the Apostles whose name they bore.
Not all Christians hold to the above early Church explanation. A summary of alternate opinions can be found in a footnote to Eusebius' Church History in Schaff's Nicene and Post-Nicene Fathers (Series 2, Volume 1):
The question as to the real authorship of our second Gospel [Mark's],
or rather as to its composition and its relation to Matthew and Luke,
is a very difficult one. The relationship of the three synoptical
Gospels was first discussed by Augustine (De Consensu
Evangelistarum), who defended the traditional order, but made Mark
dependent upon Matthew. This view prevailed until the beginning of the
present century, when the problem was attacked anew, and since then it
has been the crux of the literary criticism of the Bible. The three
have been held to be dependent upon each other, and every possible
order has found its advocates; a common source has been assumed for
the three: the Hebrew Matthew, the Gospel according to the Hebrews
(see Bk. III. chap. 25, note 24), our canonical Gospel of Mark, or an
original Mark, resembling the present one; a number of fragmentary
documents have been assumed; while others, finally, have admitted only
oral tradition as the basis. According to Baur’s tendency theory,
Matthew (polemically Jewish-Christian) came first, followed by an
original Luke (polemically Pauline-Christian), then by our Mark, which
was based upon both and written in the interest of neutrality, and
lastly by our present Luke, designed as a final irenicum. This view
now finds few advocates. The whole matter is still unsettled, but
criticism seems to be gradually converging toward a common ground type
(or rather two independent types) for all three while at the same time
maintaining the relative independence of the three, one toward the
other. What these ground types were, is a matter of still sharper
dispute, although criticism is gradually drawing their larger features
with more and more certainty and clearness. (The latest discussion
upon the subject by Handmann, das Hebräer-Evangelium, makes the two
types the “Ur-Marcus” and the Gospel of the Hebrews.) That in the last
analysis, however, some space must still be left for floating
tradition, or for documents irreducible to the one or two types, seems
absolutely certain. For further information as to the state of
discussion upon this intricate problem, see among recent works,
especially Weiss, Einleitung, p. 473 sqq., Holtzmann, Einleitung,
p. 328 sqq., and Schaff, Ch. Hist. I. 575 sqq., where the literature
down to 1882 is given with great fullness. Conservative opinion puts
the composition of all the synoptic Gospels before the destruction of
Jerusalem (for the date of Luke, see III. 4, note 12); but the
critical school, while throwing the original type back of that date,
considers the composition of our present Gospels to have been the
gradual work of years, assuming that they were not finally
crystallized into the form in which we have them before the second
century.
Some interesting points in this are that (1) Mark's Gospel began to be divorced from Peter and associated with Matthew around a century after Eusebius' history by Augustine, who (in Carthage) was somewhat removed from the theological center of Christianity at the time (Constantinople); and (2) the more exotic theories about the origin of Mark's Gospel have arisen only very recently (in Church time, anyway).